Content check — Algebra and Trig, Chapter 13: Sequences, Probability, and Counting Theory

Every problem in the chapter beside the book’s answer, a chapter at a time as you scroll.

Chapter 13: Sequences, Probability, and Counting Theory

318 problems · 21 figures

Try It #1 §13.1 source ↗

Write the first five terms of the sequence defined by the explicit formula

t_{n} = 5 n - 4.

Answer

The first five terms are

{1, 6, 11, 16, 21} .

Try It #2 §13.1 source ↗

Write the first five terms of the sequence.

a_{n} = \frac{4 n}{{(- 2)}^{n}}

Answer

The first five terms are

{- 2, 2, - \frac{3}{2}, 1, - \frac{5}{8}} .

Try It #3 §13.1 source ↗

Write the first six terms of the sequence.

a_{n} = {\begin{array}{l} 2 n^{3} & if n is odd \\ \frac{5 n}{2} & if n is even \end{array}

Answer

The first six terms are

{2, 5, 54, 10, 250, 15} .

Try It #4 §13.1 source ↗

Write an explicit formula for the $n th$ term of the sequence.

{9, - 81, 729, - 6,561, 59,049, …}

Answer

a_{n} =

{(- 1)}^{n + 1} 9^{n}

Try It #5 §13.1 source ↗

Write an explicit formula for the $n th$ term of the sequence.

{- \frac{3}{4}, - \frac{9}{8}, - \frac{27}{12}, - \frac{81}{16}, - \frac{243}{20}, ...}

Answer

a_{n} = - \frac{3^{n}}{4 n}

Try It #6 §13.1 source ↗

Write an explicit formula for the $n th$ term of the sequence.

{\frac{1}{e^{2}}, \frac{1}{e}, 1, e, e^{2}, ...}

Answer

a_{n} = e^{n - 3}

Try It #7 §13.1 source ↗

Write the first five terms of the sequence defined by the recursive formula.

\begin{array}{l} a_{1} = 2 \\ a_{n} = 2 a_{n - 1} + 1, for n \geq 2 \end{array}

Answer

{2, 5, 11, 23, 47}

Try It #8 §13.1 source ↗

Write the first 8 terms of the sequence defined by the recursive formula.

\begin{array}{l} \begin{array}{l} a_{1} = 0 \end{array} \\ a_{2} = 1 \\ a_{3} = 1 \\ a_{n} = \frac{a_{n - 1}}{a_{n - 2}} + a_{n - 3}, for n \geq 4 \end{array}

Answer

{0, 1, 1, 1, 2, 3, \frac{5}{2}, \frac{17}{6}} .

Try It #9 §13.1 source ↗

Write the first five terms of the sequence defined by the explicit formula

a_{n} =

\frac{(n + 1)!}{2 n} .

Answer

The first five terms are

{1, \frac{3}{2}, 4, 15, 72} .

13.1-1 §13.1 source ↗

Discuss the meaning of a sequence. If a finite sequence is defined by a formula, what is its domain? What about an infinite sequence?

Answer

A sequence is an ordered list of numbers that can be either finite or infinite in number. When a finite sequence is defined by a formula, its domain is a subset of the non-negative integers. When an infinite sequence is defined by a formula, its domain is all positive or all non-negative integers.

13.1-3 §13.1 source ↗

Is the ordered set of even numbers an infinite sequence? What about the ordered set of odd numbers? Explain why or why not.

Answer

Yes, both sets go on indefinitely, so they are both infinite sequences.

13.1-5 §13.1 source ↗

What is a factorial, and how is it denoted? Use an example to illustrate how factorial notation can be beneficial.

Answer

A factorial is the product of a positive integer and all the positive integers below it. An exclamation point is used to indicate the operation. Answers may vary. An example of the benefit of using factorial notation is when indicating the product It is much easier to write than it is to write out

13 \cdot 12 \cdot 11 \cdot 10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 .

13.1-7 §13.1 source ↗

For the following exercises, write the first four terms of the sequence.

$a_{n} = - \frac{16}{n + 1}$

Answer

First four terms:

-

8,

-

\frac{16}{3},

-

4,

-

\frac{16}{5}

13.1-9 §13.1 source ↗

For the following exercises, write the first four terms of the sequence.

$a_{n} = \frac{2^{n}}{n^{3}}$

Answer

First four terms:

2, \frac{1}{2}, \frac{8}{27}, \frac{1}{4}

13.1-11 §13.1 source ↗

For the following exercises, write the first four terms of the sequence.

$a_{n} =$ $1.25 \cdot {(- 4)}^{n - 1}$

Answer

First four terms:

1.25,

-

5,

20,

-

80

13.1-13 §13.1 source ↗

For the following exercises, write the first four terms of the sequence.

$a_{n} = \frac{n^{2}}{2 n + 1}$

Answer

First four terms:

\frac{1}{3},

\frac{4}{5},

\frac{9}{7},

\frac{16}{9}

13.1-15 §13.1 source ↗

For the following exercises, write the first four terms of the sequence.

$a_{n} =$ $-$ $(\frac{4 \cdot {(- 5)}^{n - 1}}{5})$

Answer

First four terms:

-

\frac{4}{5},

4,

-

20,

100

13.1-17 §13.1 source ↗

For the following exercises, write the first eight terms of the piecewise sequence.

$a_{n} = {\begin{array}{l} \frac{n^{2}}{2 n + 1} & if n \leq 5 \\ n^{2} - 5 & if n >5 \end{array}$

Answer

\frac{1}{3},

\frac{4}{5},

\frac{9}{7},

\frac{16}{9},

\frac{25}{11},

31,

44,

59

13.1-19 §13.1 source ↗

For the following exercises, write the first eight terms of the piecewise sequence.

$a_{n} = {\begin{array}{l} - 0.6 \cdot 5^{n - 1} & if n is prime or 1 \\ 2.5 \cdot {(- 2)}^{n - 1} & if n is composite \end{array}$

Answer

-

0.6,

-

3,

-

15,

-

20,

-

375,

-

80,

-

9375,

-

320

13.1-21 §13.1 source ↗

For the following exercises, write an explicit formula for each sequence.

$4,$ $7,$ $12,$ $19,$ $28,$ $\dots$

Answer

a_{n} = n^{2} + 3

13.1-23 §13.1 source ↗

For the following exercises, write an explicit formula for each sequence.

$1,$ $1,$ $\frac{4}{3},$ $2,$ $\frac{16}{5},$ $\dots$

Answer

a_{n} =

\frac{2^{n}}{2 n} or \frac{2^{n - 1}}{n}

13.1-25 §13.1 source ↗

For the following exercises, write an explicit formula for each sequence.

$1,$ $-$ $\frac{1}{2},$ $\frac{1}{4},$ $-$ $\frac{1}{8},$ $\frac{1}{16},$ $\dots$

Answer

a_{n} = {(- \frac{1}{2})}^{n - 1}

13.1-27 §13.1 source ↗

For the following exercises, write the first five terms of the sequence.

$a_{1} =$ $3,$ $a_{n} =$ $(- 3) a_{n - 1}$

Answer

First five terms:

3,

-

9,

27,

-

81,

243

13.1-29 §13.1 source ↗

For the following exercises, write the first five terms of the sequence.

$a_{1} =$ $-$ $1,$ $a_{n} =$ $\frac{{(- 3)}^{n - 1}}{a_{n - 1} - 2}$

Answer

First five terms:

-

1,

1,

-

9,

\frac{27}{11},

\frac{891}{5}

13.1-31 §13.1 source ↗

For the following exercises, write the first eight terms of the sequence.

$a_{1} =$ $\frac{1}{24},$ $a_{2} =$ $1,$ $a_{n} =$ $(2 a_{n - 2}) (3 a_{n - 1})$

Answer

\frac{1}{24},

1, \frac{1}{4},

\frac{3}{2},

\frac{9}{4},

\frac{81}{4},

\frac{2187}{8},

\frac{531, 441}{16}

13.1-33 §13.1 source ↗

For the following exercises, write the first eight terms of the sequence.

$a_{1} =$ $2,$ $a_{2} =$ $10,$ $a_{n} =$ $\frac{2 (a_{n - 1} + 2)}{a_{n - 2}}$

Answer

2,

10,

12,

\frac{14}{5},

\frac{4}{5},

2,

10,

12

13.1-35 §13.1 source ↗

For the following exercises, write a recursive formula for each sequence.

$-$ $8,$ $-$ $6,$ $-$ $3,$ $1,$ $6,$ $\dots$

Answer

a_{1} =

-

8,

a_{n} =

a_{n - 1} +

n

13.1-37 §13.1 source ↗

For the following exercises, write a recursive formula for each sequence.

$35,$ $38,$ $41,$ $44,$ $47,$ $\dots$

Answer

a_{1} =

35,

a_{n} =

a_{n - 1} +

3

13.1-39 §13.1 source ↗

For the following exercises, evaluate the factorial.

$6!$

Answer

720

13.1-41 §13.1 source ↗

For the following exercises, evaluate the factorial.

$\frac{12!}{6!}$

Answer

665, 280

13.1-43 §13.1 source ↗

For the following exercises, write the first four terms of the sequence.

$a_{n} = \frac{n!}{n^{2}}$

Answer

First four terms:

1, \frac{1}{2}, \frac{2}{3}, \frac{3}{2}

13.1-45 §13.1 source ↗

For the following exercises, write the first four terms of the sequence.

$a_{n} = \frac{n!}{n^{2} - n - 1}$

Answer

First four terms:

-

1,

2,

\frac{6}{5},

\frac{24}{11}

13.1-47 §13.1 source ↗

For the following exercises, graph the first five terms of the indicated sequence

$a_{n} = \frac{{(- 1)}^{n}}{n} + n$

Answer

Graph of a scattered plot with points at (1, 0), (2, 5/2), (3, 8/3), (4, 17/4), and (5, 24/5). The x-axis is labeled n and the y-axis is labeled a_n.

13.1-49 §13.1 source ↗

For the following exercises, graph the first five terms of the indicated sequence

$a_{1} =$ $2,$ $a_{n} =$ ${(- a_{n - 1} + 1)}^{2}$

Answer

Graph of a scattered plot with points at (1, 2), (2, 1), (3, 0), (4, 1), and (5, 0). The x-axis is labeled n and the y-axis is labeled a_n.

13.1-51 §13.1 source ↗

For the following exercises, graph the first five terms of the indicated sequence

$a_{n} =$ $\frac{(n + 1)!}{(n - 1)!}$

Answer

Graph of a scattered plot with labeled points: (1, 2), (2, 6), (3, 12), (4, 20), and (5, 30). The x-axis is labeled n and the y-axis is labeled a_n.

13.1-53 §13.1 source ↗

For the following exercises, write an explicit formula for the sequence using the first five points shown on the graph.

Graph of a scattered plot with labeled points: (1, 0.5), (2, 1), (3, 2), (4, 4), and (5, 8). The x-axis is labeled n and the y-axis is labeled a_n.

Answer

a_{n} = 2^{n - 2}

13.1-55 §13.1 source ↗

For the following exercises, write a recursive formula for the sequence using the first five points shown on the graph.

Graph of a scattered plot with labeled points: (1, 6), (2, 7), (3, 9), (4, 13), and (5, 21). The x-axis is labeled n and the y-axis is labeled a_n.

Answer

a_{1} =

6,

a_{n} =

2 a_{n - 1} -

5

13.1-57 §13.1 source ↗

Follow these steps to evaluate a sequence defined recursively using a graphing calculator:

For the following exercises, use the steps above to find the indicated term or terms for the sequence.

Find the first five terms of the sequence $a_{1} = \frac{87}{111}$ , $a_{n} =$ $\frac{4}{3} a_{n - 1} +$ $\frac{12}{37} .$ Use the >Frac feature to give fractional results.

Answer

First five terms:

\frac{29}{37}

\frac{152}{111}

\frac{716}{333}

\frac{3188}{999}

\frac{13724}{2997}

13.1-59 §13.1 source ↗

Follow these steps to evaluate a sequence defined recursively using a graphing calculator:

For the following exercises, use the steps above to find the indicated term or terms for the sequence.

Find the first five terms of the sequence $a_{1} = 2$ , $a_{n} =$ $2^{[(a_{n} - 1) - 1]} +$ $1.$

Answer

First five terms: 2, 3, 5, 17, 65537

13.1-61 §13.1 source ↗

Follow these steps to evaluate a sequence defined recursively using a graphing calculator:

For the following exercises, use the steps above to find the indicated term or terms for the sequence.

Find the tenth term of the sequence $a_{1} = 2$ , $a_{n} = n a_{n - 1}$

Answer

a_{10} =

7,

257,

600

13.1-63 §13.1 source ↗

Follow these steps to evaluate a finite sequence defined by an explicit formula. Using a TI-84, do the following.

Using a TI-83, do the following.

For the following exercises, use the steps above to find the indicated terms for the sequence. Round to the nearest thousandth when necessary.

List the first six terms of the sequence $a_{n} =$ $\frac{n^{3} - 3.5 n^{2} + 4.1 n - 1.5}{2.4 n} .$

Answer

First six terms: 0.042, 0.146, 0.875, 2.385, 4.708

13.1-65 §13.1 source ↗

Follow these steps to evaluate a finite sequence defined by an explicit formula. Using a TI-84, do the following.

Using a TI-83, do the following.

For the following exercises, use the steps above to find the indicated terms for the sequence. Round to the nearest thousandth when necessary.

List the first four terms of the sequence $a_{n} =$ ${5.7}^{n} +$ $0.275 (n - 1)!$

Answer

First four terms: 5.975, 2.765, 185.743, 1057.25, 6023.521

13.1-67 §13.1 source ↗

Follow these steps to evaluate a finite sequence defined by an explicit formula. Using a TI-84, do the following.

Using a TI-83, do the following.

For the following exercises, use the steps above to find the indicated terms for the sequence. Round to the nearest thousandth when necessary.

Consider the sequence defined by $a_{n} = - 6 - 8 n .$ Is $a_{n} = - 421$ a term in the sequence? Verify the result.

Answer

a_{n} = - 421

is a term in the sequence, then solving the equation

- 421 = - 6 - 8 n

for

n

will yield a non-negative integer. However, if

- 421 = - 6 - 8 n,

then

n = 51.875

a_{n} = - 421

is not a term in the sequence.

13.1-69 §13.1 source ↗

Follow these steps to evaluate a finite sequence defined by an explicit formula. Using a TI-84, do the following.

Using a TI-83, do the following.

For the following exercises, use the steps above to find the indicated terms for the sequence. Round to the nearest thousandth when necessary.

Find a recursive formula for the sequence 1, 0, −1, −1, 0, 1, 1, 0, −1, −1, 0, 1, 1, .... (Hint: find a pattern for $a_{n}$ based on the first two terms.)

Answer

a_{1} =

1,

a_{2} =

0,

a_{n} =

a_{n - 1} -

a_{n - 2}

13.1-71 §13.1 source ↗

Follow these steps to evaluate a finite sequence defined by an explicit formula. Using a TI-84, do the following.

Using a TI-83, do the following.

For the following exercises, use the steps above to find the indicated terms for the sequence. Round to the nearest thousandth when necessary.

Prove the conjecture made in the preceding exercise.

Answer

\frac{(n + 2)!}{(n - 1)!} =

\frac{(n + 2) \cdot (n + 1) \cdot (n) \cdot (n - 1) \cdot ... \cdot 3 \cdot 2 \cdot 1}{(n - 1) \cdot ... \cdot 3 \cdot 2 \cdot 1} =

n (n + 1) (n + 2) =

n^{3} +

3 n^{2} +

2 n

Try It #1 §13.2 source ↗

Is the given sequence arithmetic? If so, find the common difference.

{18, 16, 14, 12, 10, \dots}

Answer

The sequence is arithmetic. The common difference is

- 2.

Try It #2 §13.2 source ↗

Is the given sequence arithmetic? If so, find the common difference.

{1, 3, 6, 10, 15, \dots}

Answer

The sequence is not arithmetic because

3 - 1 \neq 6 - 3.

Try It #3 §13.2 source ↗

List the first five terms of the arithmetic sequence with

a_{1} = 1

and

d = 5

Answer

{1, 6, 11, 16, 21}

Try It #4 §13.2 source ↗

Given

a_{3} = 7

and

a_{5} = 17

, find

a_{2}

Answer

a_{2} = 2

Try It #5 §13.2 source ↗

Write a recursive formula for the arithmetic sequence.

{25, 37, 49, 61, …}

Answer

\begin{array}{l} a_{1} = 25 \\ a_{n} = a_{n - 1} + 12, for n \geq 2 \end{array}

Try It #6 §13.2 source ↗

Write an explicit formula for the following arithmetic sequence.

{50, 47, 44, 41, \dots}

Answer

a_{n} = 53 - 3 n

Try It #7 §13.2 source ↗

Find the number of terms in the finite arithmetic sequence.

{6, 11, 16, ..., 56}

Answer

There are 11 terms in the sequence.

Try It #8 §13.2 source ↗

A woman decides to go for a 10-minute run every day this week and plans to increase the time of her daily run by 4 minutes each week. Write a formula for the time of her run after n weeks. How long will her daily run be 8 weeks from today?

Answer

The formula is

T_{n} = 10 + 4 n,

and it will take her 42 minutes.

13.2-1 §13.2 source ↗

What is an arithmetic sequence?

Answer

A sequence where each successive term of the sequence increases (or decreases) by a constant value.

13.2-3 §13.2 source ↗

How do we determine whether a sequence is arithmetic?

Answer

We find whether the difference between all consecutive terms is the same. This is the same as saying that the sequence has a common difference.

13.2-5 §13.2 source ↗

Describe how linear functions and arithmetic sequences are similar. How are they different?

Answer

Both arithmetic sequences and linear functions have a constant rate of change. They are different because their domains are not the same; linear functions are defined for all real numbers, and arithmetic sequences are defined for natural numbers or a subset of the natural numbers.

13.2-7 §13.2 source ↗

For the following exercises, find the common difference for the arithmetic sequence provided.

${0, \frac{1}{2}, 1, \frac{3}{2}, 2, ...}$

Answer

The common difference is

\frac{1}{2}

13.2-9 §13.2 source ↗

For the following exercises, determine whether the sequence is arithmetic. If so find the common difference.

${4, 16, 64, 256, 1024, ...}$

Answer

The sequence is not arithmetic because

16 - 4 \neq 64 - 16.

13.2-11 §13.2 source ↗

For the following exercises, write the first five terms of the arithmetic sequence given the first term and common difference.

$a_{1} = 0$ , $d = \frac{2}{3}$

Answer

0,

\frac{2}{3},

\frac{4}{3},

2,

\frac{8}{3}

13.2-13 §13.2 source ↗

For the following exercises, write the first five terms of the arithmetic series given two terms.

$a_{13} =$ $-$ $60,$ $a_{33} =$ $-$ $160$

Answer

0,

-

5,

-

10,

-

15,

-

20

13.2-15 §13.2 source ↗

For the following exercises, find the specified term for the arithmetic sequence given the first term and common difference.

First term is 4, common difference is 5, find the 4^th term.

Answer

a_{4} = 19

13.2-17 §13.2 source ↗

For the following exercises, find the specified term for the arithmetic sequence given the first term and common difference.

First term is 6, common difference is 7, find the 6^th term.

Answer

a_{6} = 41

13.2-19 §13.2 source ↗

For the following exercises, find the first term given two terms from an arithmetic sequence.

Find the first term or $a_{1}$ of an arithmetic sequence if $a_{6} = 12$ and $a_{14} = 28.$

Answer

a_{1} = 2

13.2-21 §13.2 source ↗

For the following exercises, find the first term given two terms from an arithmetic sequence.

Find the first term or $a_{1}$ of an arithmetic sequence if $a_{8} = 40$ and $a_{23} = 115.$

Answer

a_{1} = 5

13.2-23 §13.2 source ↗

For the following exercises, find the first term given two terms from an arithmetic sequence.

Find the first term or $a_{1}$ of an arithmetic sequence if $a_{11} = 11$ and $a_{21} = 16.$

Answer

a_{1} = 6

13.2-25 §13.2 source ↗

For the following exercises, find the specified term given two terms from an arithmetic sequence.

$a_{3} = - 17.1$ and $a_{10} = - 15.7.$ Find $a_{21} .$

Answer

a_{21} = - 13.5

13.2-27 §13.2 source ↗

For the following exercises, use the recursive formula to write the first five terms of the arithmetic sequence.

$a_{1} =$ $-$ $19;$ $a_{n} =$ $a_{n - 1} -$ $1.4$

Answer

-

19,

-

20.4,

-

21.8,

-

23.2,

-

24.6

13.2-29 §13.2 source ↗

For the following exercises, write a recursive formula for each arithmetic sequence.

$a =$ ${17, 26, 35, ...}$

Answer

\begin{array}{l} a_{1} = 17; a_{n} = a_{n - 1} + 9 & n \geq 2 \end{array}

13.2-31 §13.2 source ↗

For the following exercises, write a recursive formula for each arithmetic sequence.

$a =$ ${12, 17, 22, ...}$

Answer

\begin{array}{l} a_{1} = 12; a_{n} = a_{n - 1} + 5 & n \geq 2 \end{array}

13.2-33 §13.2 source ↗

For the following exercises, write a recursive formula for each arithmetic sequence.

$a =$ ${8.9, 10.3, 11.7, ...}$

Answer

\begin{array}{l} a_{1} = 8.9; a_{n} = a_{n - 1} + 1.4 & n \geq 2 \end{array}

13.2-35 §13.2 source ↗

For the following exercises, write a recursive formula for each arithmetic sequence.

$a =$ ${\frac{1}{5}, \frac{9}{20}, \frac{7}{10}, ...}$

Answer

\begin{array}{l} a_{1} = \frac{1}{5}; a_{n} = a_{n - 1} + \frac{1}{4} & n \geq 2 \end{array}

13.2-37 §13.2 source ↗

For the following exercises, write a recursive formula for each arithmetic sequence.

$a =$ ${\frac{1}{6}, - \frac{11}{12}, - 2, ...}$

Answer

\begin{array}{l} _{1} = \frac{1}{6}; a_{n} = a_{n - 1} - \frac{13}{12} & n \geq 2 \end{array}

13.2-39 §13.2 source ↗

For the following exercises, write a recursive formula for the given arithmetic sequence, and then find the specified term.

$a =$ ${4, 11, 18, ...};$ Find the 14^th term.

Answer

a_{1} =

4;

a_{n} =

a_{n - 1} +

7;

a_{14} =

95

13.2-41 §13.2 source ↗

For the following exercises, use the explicit formula to write the first five terms of the arithmetic sequence.

$a_{n} = 24 - 4 n$

Answer

First five terms:

20,

16,

12,

8,

4.

13.2-43 §13.2 source ↗

For the following exercises, write an explicit formula for each arithmetic sequence.

$a =$ ${3, 5, 7, ...}$

Answer

a_{n} = 1 + 2 n

13.2-45 §13.2 source ↗

For the following exercises, write an explicit formula for each arithmetic sequence.

$a =$ ${- 5, 95, 195, ...}$

Answer

a_{n} =

-

105 +

100 n

13.2-47 §13.2 source ↗

For the following exercises, write an explicit formula for each arithmetic sequence.

$a =$ ${1.8, 3.6, 5.4, ...}$

Answer

a_{n} = 1.8 n

13.2-49 §13.2 source ↗

For the following exercises, write an explicit formula for each arithmetic sequence.

$a =$ ${15.8, 18.5, 21.2, ...}$

Answer

a_{n} =

13.1 +

2.7 n

13.2-51 §13.2 source ↗

For the following exercises, write an explicit formula for each arithmetic sequence.

$a =$ ${0, \frac{1}{3}, \frac{2}{3}, ...}$

Answer

a_{n} = \frac{1}{3} n - \frac{1}{3}

13.2-53 §13.2 source ↗

For the following exercises, find the number of terms in the given finite arithmetic sequence.

$a =$ ${3, - 4, - 11, ..., - 60}$

Answer

There are 10 terms in the sequence.

13.2-55 §13.2 source ↗

For the following exercises, find the number of terms in the given finite arithmetic sequence.

$a =$ ${\frac{1}{2}, 2, \frac{7}{2}, ..., 8}$

Answer

There are 6 terms in the sequence.

13.2-57 §13.2 source ↗

For the following exercises, determine whether the graph shown represents an arithmetic sequence.

Graph of a scattered plot with labeled points: (1, 1.5), (2, 2.25), (3, 3.375), (4, 5.0625), and (5, 7.5938). The x-axis is labeled n and the y-axis is labeled a_n.

Answer

The graph does not represent an arithmetic sequence.

13.2-59 §13.2 source ↗

For the following exercises, use the information provided to graph the first 5 terms of the arithmetic sequence.

$a_{1} =$ $9;$ $a_{n} =$ $a_{n - 1} -$ $10$

Answer

Graph of a scattered plot with labeled points: (1, 9), (2, -1), (3, -11), (4, -21), and (5, -31). The x-axis is labeled n and the y-axis is labeled a_n.

13.2-61 §13.2 source ↗

For the following exercises, follow the steps to work with the arithmetic sequence $a_{n} = 3 n - 2$ using a graphing calculator:

What are the first seven terms shown in the column with the heading $u (n) ?$

Answer

1,

4,

7,

10,

13,

16,

19

13.2-63 §13.2 source ↗

For the following exercises, follow the steps to work with the arithmetic sequence $a_{n} = 3 n - 2$ using a graphing calculator:

Press [WINDOW]. Set $n Min = 1$ , $n Max = 5$ , $x Min = 0$ , $x Max = 6$ , $y Min = - 1$ , and $y Max = 14.$ Then press [GRAPH]. Graph the sequence as it appears on the graphing calculator.

Answer

Graph of a scattered plot with labeled points: (1, 1), (2, 4), (3, 7), (4, 10), and (5, 13). The x-axis is labeled n and the y-axis is labeled a_n.

13.2-65 §13.2 source ↗

For the following exercises, follow the steps given above to work with the arithmetic sequence $a_{n} = \frac{1}{2} n + 5$ using a graphing calculator.

Graph the sequence as it appears on the graphing calculator. Be sure to adjust the WINDOW settings as needed.

Answer

Graph of a scattered plot with labeled points: (1, 5.5), (2, 6), (3, 6.5), (4, 7), and (5, 7.5). The x-axis is labeled n and the y-axis is labeled a_n.

13.2-67 §13.2 source ↗

For the following exercises, follow the steps given above to work with the arithmetic sequence $a_{n} = \frac{1}{2} n + 5$ using a graphing calculator.

Give two examples of arithmetic sequences whose 10^th terms are $206.$

Answer

Answers will vary. Examples:

a_{n} = 20.6 n

and

a_{n} = 2 + 20.4 n.

13.2-69 §13.2 source ↗

For the following exercises, follow the steps given above to work with the arithmetic sequence $a_{n} = \frac{1}{2} n + 5$ using a graphing calculator.

Find the 11^th term of the arithmetic sequence ${3 a - 2 b, a + 2 b, - a + 6 b \dots} .$

Answer

a_{11} =

-

17 a +

38 b

13.2-71 §13.2 source ↗

For the following exercises, follow the steps given above to work with the arithmetic sequence $a_{n} = \frac{1}{2} n + 5$ using a graphing calculator.

At which term does the sequence ${\frac{17}{3}, \frac{31}{6}, \frac{14}{3}, ...}$ begin to have negative values?

Answer

The sequence begins to have negative values at the 13^th term,

a_{13} = - \frac{1}{3}

13.2-73 §13.2 source ↗

For the following exercises, follow the steps given above to work with the arithmetic sequence $a_{n} = \frac{1}{2} n + 5$ using a graphing calculator.

Write an arithmetic sequence using a recursive formula. Show the first 4 terms, and then find the 31^st term.

Answer

Answers will vary. Check to see that the sequence is arithmetic. Example: Recursive formula:

a_{1} =

3,

a_{n} =

a_{n - 1} -

3.

First 4 terms:

\begin{array}{l} 3, 0, - 3, - 6 & a_{31} = - 87 \end{array}

Try It #1 §13.3 source ↗

Is the sequence geometric? If so, find the common ratio.

5, 10, 15, 20, ...

Answer

The sequence is not geometric because

\frac{10}{5} \neq \frac{15}{10}

Try It #2 §13.3 source ↗

Is the sequence geometric? If so, find the common ratio.

100, 20, 4, \frac{4}{5}, ...

Answer

The sequence is geometric. The common ratio is

\frac{1}{5}

Try It #3 §13.3 source ↗

List the first five terms of the geometric sequence with

a_{1} = 18

and

r = \frac{1}{3} .

Answer

{18, 6, 2, \frac{2}{3}, \frac{2}{9}}

Try It #4 §13.3 source ↗

Write a recursive formula for the following geometric sequence.

{2, \frac{4}{3}, \frac{8}{9}, \frac{16}{27}, ...}

Answer

\begin{array}{l} a_{1} = 2 \\ a_{n} = \frac{2}{3} a_{n - 1} for n \geq 2 \end{array}

Try It #5 §13.3 source ↗

Given a geometric sequence with

a_{2} = 4

and

a_{3} = 32

, find

a_{6} .

Answer

a_{6} = 16, 384

Try It #6 §13.3 source ↗

Write an explicit formula for the following geometric sequence.

{–1, 3, –9, 27, ...}

Answer

a_{n} = - {(- 3)}^{n - 1}

Try It #7 §13.3 source ↗

A business starts a new website. Initially the number of hits is 293 due to the curiosity factor. The business estimates the number of hits will increase by 2.6% per week.

ⓐWrite a formula for the number of hits.

ⓑEstimate the number of hits in 5 weeks.

Answer

ⓐ $P_{n} =$ $293 \cdot 1.026 a^{n}$

ⓑThe number of hits will be about 333.

13.3-1 §13.3 source ↗

What is a geometric sequence?

Answer

A sequence in which the ratio between any two consecutive terms is constant.

13.3-3 §13.3 source ↗

What is the procedure for determining whether a sequence is geometric?

Answer

Divide each term in a sequence by the preceding term. If the resulting quotients are equal, then the sequence is geometric.

13.3-5 §13.3 source ↗

Describe how exponential functions and geometric sequences are similar. How are they different?

Answer

Both geometric sequences and exponential functions have a constant ratio. However, their domains are not the same. Exponential functions are defined for all real numbers, and geometric sequences are defined only for positive integers. Another difference is that the base of a geometric sequence (the common ratio) can be negative, but the base of an exponential function must be positive.

13.3-7 §13.3 source ↗

For the following exercises, find the common ratio for the geometric sequence.

$-$ $0.125,$ $0.25,$ $-$ $0.5,$ $1,$ $-$ $2,$ $...$

Answer

The common ratio is

- 2

13.3-9 §13.3 source ↗

For the following exercises, determine whether the sequence is geometric. If so, find the common ratio.

$-$ $6,$ $-$ $12,$ $-$ $24,$ $-$ $48,$ $-$ $96,$ $...$

Answer

The sequence is geometric. The common ratio is 2.

13.3-11 §13.3 source ↗

For the following exercises, determine whether the sequence is geometric. If so, find the common ratio.

$-$ $1,$ $\frac{1}{2},$ $-$ $\frac{1}{4},$ $\frac{1}{8},$ $-$ $\frac{1}{16},$ $...$

Answer

The sequence is geometric. The common ratio is

- \frac{1}{2} .

13.3-13 §13.3 source ↗

For the following exercises, determine whether the sequence is geometric. If so, find the common ratio.

$0.8,$ $4,$ $20,$ $100,$ $500,$ $...$

Answer

The sequence is geometric. The common ratio is

5.

13.3-15 §13.3 source ↗

For the following exercises, write the first five terms of the geometric sequence, given the first term and common ratio.

$\begin{matrix} a_{1} = 5, & r = \frac{1}{5} \end{matrix}$

Answer

5,

1,

\frac{1}{5},

\frac{1}{25},

\frac{1}{125}

13.3-17 §13.3 source ↗

For the following exercises, write the first five terms of the geometric sequence, given any two terms.

$\begin{matrix} a_{6} = 25, & a_{8} \end{matrix} = 6.25$

Answer

800,

400,

200,

100,

50

13.3-19 §13.3 source ↗

For the following exercises, find the specified term for the geometric sequence, given the first term and common ratio.

The first term is 16 and the common ratio is $- \frac{1}{3} .$ Find the 4^th term.

Answer

a_{4} = - \frac{16}{27}

13.3-21 §13.3 source ↗

For the following exercises, find the specified term for the geometric sequence, given the first four terms.

$a_{n} =$ ${- 2, \frac{2}{3}, - \frac{2}{9}, \frac{2}{27}, ...} .$ Find $a_{7} .$

Answer

a_{7} = - \frac{2}{729}

13.3-23 §13.3 source ↗

For the following exercises, write the first five terms of the geometric sequence.

$\begin{matrix} a_{1} = 7, & a_{n} = 0.2 a_{n - 1} \end{matrix}$

Answer

7,

1.4,

0.28,

0.056,

0.0112

13.3-25 §13.3 source ↗

For the following exercises, write a recursive formula for each geometric sequence.

$a_{n} =$ ${- 32, - 16, - 8, - 4, ...}$

Answer

\begin{matrix} a = 1 - 32, & a_{n} = \frac{1}{2} a_{n - 1} \end{matrix}

13.3-27 §13.3 source ↗

For the following exercises, write a recursive formula for each geometric sequence.

$a_{n} =$ ${10, - 3, 0.9, - 0.27, ...}$

Answer

\begin{matrix} a_{1} = 10, & a_{n} = - 0.3 a_{n - 1} \end{matrix}

13.3-29 §13.3 source ↗

For the following exercises, write a recursive formula for each geometric sequence.

$a_{n} =$ ${\frac{3}{5}, \frac{1}{10}, \frac{1}{60}, \frac{1}{360}, ...}$

Answer

\begin{matrix} a_{1} = \frac{3}{5}, & a_{n} = \frac{1}{6} a_{n - 1} \end{matrix}

13.3-31 §13.3 source ↗

For the following exercises, write a recursive formula for each geometric sequence.

$a_{n} =$ ${\frac{1}{512}, - \frac{1}{128}, \frac{1}{32}, - \frac{1}{8}, ...}$

Answer

a_{1} =

\frac{1}{512},

a_{n} =

-

4 a_{n - 1}

13.3-33 §13.3 source ↗

For the following exercises, write the first five terms of the geometric sequence.

$a_{n} =$ $12 \cdot {(- \frac{1}{2})}^{n - 1}$

Answer

12,

-

6,

3,

-

\frac{3}{2},

\frac{3}{4}

13.3-35 §13.3 source ↗

For the following exercises, write an explicit formula for each geometric sequence.

$a_{n} =$ ${1, 3, 9, 27, ...}$

Answer

a_{n} = 3^{n - 1}

13.3-37 §13.3 source ↗

For the following exercises, write an explicit formula for each geometric sequence.

$a_{n} =$ ${0.8, - 4, 20, - 100, ...}$

Answer

a_{n} =

0.8 \cdot {(- 5)}^{n - 1}

13.3-39 §13.3 source ↗

For the following exercises, write an explicit formula for each geometric sequence.

$a_{n} =$ ${- 1, - \frac{4}{5}, - \frac{16}{25}, - \frac{64}{125}, ...}$

Answer

a_{n} = - {(\frac{4}{5})}^{n - 1}

13.3-41 §13.3 source ↗

For the following exercises, write an explicit formula for each geometric sequence.

$a_{n} =$ ${3, - 1, \frac{1}{3}, - \frac{1}{9}, ...}$

Answer

a_{n} =

3 \cdot {(- \frac{1}{3})}^{n - 1}

13.3-43 §13.3 source ↗

For the following exercises, find the specified term for the geometric sequence given.

Let $a_{n} =$ $-$ ${(- \frac{1}{3})}^{n - 1} .$ Find $a_{12} .$

Answer

a_{12} =

\frac{1}{177, 147}

13.3-45 §13.3 source ↗

For the following exercises, find the number of terms in the given finite geometric sequence.

$a_{n} =$ ${2, 1, \frac{1}{2}, ..., \frac{1}{1024}}$

Answer

There are

12

terms in the sequence.

13.3-47 §13.3 source ↗

For the following exercises, determine whether the graph shown represents a geometric sequence.

Graph of a scattered plot with labeled points: (1, -0.5), (2, 0.25), (3, 1.375), (4, 3.0625), and (5, 5.5938). The x-axis is labeled n and the y-axis is labeled a_n.

Answer

The graph does not represent a geometric sequence.

13.3-49 §13.3 source ↗

For the following exercises, use the information provided to graph the first five terms of the geometric sequence.

$\begin{matrix} a_{1} = 3, & a_{n} = 2 a_{n - 1} \end{matrix}$

Answer

Graph of a scattered plot with labeled points: (1, 3), (2, 6), (3, 12), (4, 24), and (5, 48). The x-axis is labeled n and the y-axis is labeled a_n.

13.3-51 §13.3 source ↗

For the following exercises, use the information provided to graph the first five terms of the geometric sequence.

Use recursive formulas to give two examples of geometric sequences whose 3^rd terms are $200.$

Answer

Answers will vary. Examples:

{\begin{matrix} a_{1} = 800, & a_{n} = 0.5 a \end{matrix}}_{n - 1}

and

{\begin{matrix} a_{1} = 12.5, & a_{n} = 4 a \end{matrix}}_{n - 1}

13.3-53 §13.3 source ↗

For the following exercises, use the information provided to graph the first five terms of the geometric sequence.

Find the 5^th term of the geometric sequence ${b, 4 b, 16 b, ...} .$

Answer

a_{5} = 256 b

13.3-55 §13.3 source ↗

For the following exercises, use the information provided to graph the first five terms of the geometric sequence.

At which term does the sequence ${10, 12, 14.4, 17.28, ...}$ exceed $100 ?$

Answer

The sequence exceeds

100

at the 14^th term,

a_{14} \approx 107.

13.3-57 §13.3 source ↗

For the following exercises, use the information provided to graph the first five terms of the geometric sequence.

For which term does the geometric sequence $a_{_{n}} =$ $-$ $36 {(\frac{2}{3})}^{n - 1}$ first have a non-integer value?

Answer

a_{4} = - \frac{32}{3}

is the first non-integer value

13.3-59 §13.3 source ↗

For the following exercises, use the information provided to graph the first five terms of the geometric sequence.

Use the explicit formula to write a geometric sequence whose common ratio is a decimal number between 0 and 1. Show the first 4 terms, and then find the 8^th term.

Answer

Answers will vary. Example: Explicit formula with a decimal common ratio:

a_{n} =

400 \cdot {0.5}^{n - 1};

First 4 terms:

\begin{matrix} 400, 200, 100, 50; & a_{8} = 3.125 \end{matrix}

Try It #1 §13.4 source ↗

Evaluate

\sum_{k = 2}^{5} (3 k - 1) .

Answer

Try It #2 §13.4 source ↗

Use the formula to find the sum of each arithmetic series.

$1 .4 + 1 .6 + 1 .8 + 2 .0 + 2 .2 + 2 .4 + 2 .6 + 2 .8 + 3 .0 + 3 .2 + 3 .4$

Answer

26 .4

Try It #3 §13.4 source ↗

Use the formula to find the sum of each arithmetic series.

$13 + 21 + 29 + \dots + 69$

Answer

328

Try It #4 §13.4 source ↗

Use the formula to find the sum of each arithmetic series.

$\sum_{k = 1}^{10} 5 - 6 k$

Answer

−280

Try It #5 §13.4 source ↗

A man earns $100 in the first week of June. Each week, he earns $12.50 more than the previous week. After 12 weeks, how much has he earned?

Answer

$2,025

Try It #6 §13.4 source ↗

Use the formula to find the indicated partial sum of each geometric series.

$S_{20}$ for the series $1,000 + 500 + 250 + \dots$

Answer

\approx 2, 000.00

Try It #7 §13.4 source ↗

Use the formula to find the indicated partial sum of each geometric series.

$\sum_{k = 1}^{8} 3^{k}$

Answer

9,840

Try It #8 §13.4 source ↗

At a new job, an employee’s starting salary is $32,100. She receives a 2% annual raise. How much will she have earned by the end of 8 years?

Answer

$275,513.31

Try It #9 §13.4 source ↗

Determine whether the sum of the infinite series is defined.

$\frac{1}{3} +$ $\frac{1}{2} +$ $\frac{3}{4} +$ $\frac{9}{8} +$ $...$

Answer

The sum is not defined.

Try It #10 §13.4 source ↗

Determine whether the sum of the infinite series is defined.

$24 +$ $(−12) +$ $6 +$ $(−3) +$ $...$

Answer

The sum of the infinite series is defined.

Try It #11 §13.4 source ↗

Determine whether the sum of the infinite series is defined.

$\sum_{k = 1}^{\infty} 15 \cdot {(- 0.3)}^{k}$

Answer

The sum of the infinite series is defined.

Try It #12 §13.4 source ↗

Find the sum, if it exists.

$2 + \frac{2}{3} + \frac{2}{9} + ...$

Answer

Try It #13 §13.4 source ↗

Find the sum, if it exists.

$\sum_{k = 1}^{\infty} 0.76 k + 1$

Answer

The series is not geometric.

Try It #14 §13.4 source ↗

Find the sum, if it exists.

$\sum_{k = 1}^{\infty} {(- \frac{3}{8})}^{k}$

Answer

- \frac{3}{11}

Try It #15 §13.4 source ↗

At the beginning of each month, $200 is deposited into a retirement fund. The fund earns 6% annual interest, compounded monthly, and paid into the account at the end of the month. How much is in the account if deposits are made for 10 years?

Answer

$32,775.87

13.4-1 §13.4 source ↗

What is an

n th

partial sum?

Answer

n th

partial sum is the sum of the first

n

terms of a sequence.

13.4-3 §13.4 source ↗

What is a geometric series?

Answer

A geometric series is the sum of the terms in a geometric sequence.

13.4-5 §13.4 source ↗

What is an annuity?

Answer

An annuity is a series of regular equal payments that earn a constant compounded interest.

13.4-7 §13.4 source ↗

For the following exercises, express each description of a sum using summation notation.

The sum from of $n = 0$ to $n = 4$ of $5 n$

Answer

\sum_{n = 0}^{4} 5 n

13.4-9 §13.4 source ↗

For the following exercises, express each description of a sum using summation notation.

The sum that results from adding the number 4 five times

Answer

\sum_{k = 1}^{5} 4

13.4-11 §13.4 source ↗

For the following exercises, express each arithmetic sum using summation notation.

$10 +$ $18 +$ $26 +$ $\dots +$ $162$

Answer

\sum_{k = 1}^{20} 8 k + 2

13.4-13 §13.4 source ↗

For the following exercises, use the formula for the sum of the first $n$ terms of each arithmetic sequence.

$\frac{3}{2} +$ $2 +$ $\frac{5}{2} +$ $3 +$ $\frac{7}{2}$

Answer

S_{5} =

\frac{5 (\frac{3}{2} + \frac{7}{2})}{2}

13.4-15 §13.4 source ↗

For the following exercises, use the formula for the sum of the first $n$ terms of each arithmetic sequence.

$3.2 +$ $3.4 +$ $3.6 +$ $\dots +$ $5.6$

Answer

S_{13} =

\frac{13 (3.2 + 5.6)}{2}

13.4-17 §13.4 source ↗

For the following exercises, express each geometric sum using summation notation.

$8 +$ $4 +$ $2 +$ $\dots +$ $0.125$

Answer

\sum_{k = 1}^{7} 8 \cdot {0.5}^{k - 1}

13.4-19 §13.4 source ↗

For the following exercises, use the formula for the sum of the first $n$ terms of each geometric sequence, and then state the indicated sum.

$9 + 3 + 1 + \frac{1}{3} + \frac{1}{9}$

Answer

S_{5} =

\frac{9 (1 - {(\frac{1}{3})}^{5})}{1 - \frac{1}{3}} =

\frac{121}{9} \approx

13.44

13.4-21 §13.4 source ↗

For the following exercises, use the formula for the sum of the first $n$ terms of each geometric sequence, and then state the indicated sum.

$\sum_{a = 1}^{11} 64 \cdot {0.2}^{a - 1}$

Answer

S_{11} =

\frac{64 (1 - {0.2}^{11})}{1 - 0.2} =

\frac{781, 249, 984}{9, 765, 625} \approx

80

13.4-23 §13.4 source ↗

For the following exercises, determine whether the infinite series has a sum. If so, write the formula for the sum. If not, state the reason.

$2 +$ $1.6 +$ $1.28 +$ $1.024 +$ $...$

Answer

The series is defined.

S = \frac{2}{1 - 0.8}

13.4-25 §13.4 source ↗

For the following exercises, determine whether the infinite series has a sum. If so, write the formula for the sum. If not, state the reason.

${\sum^{}}_{\infty}^{k = 1} -$ ${(- \frac{1}{2})}^{k - 1}$

Answer

The series is defined.

S = \frac{- 1}{1 - (- \frac{1}{2})}

13.4-27 §13.4 source ↗

For the following exercises, use the following scenario. Javier makes monthly deposits into a savings account. He opened the account with an initial deposit of $50. Each month thereafter he increased the previous deposit amount by $20.

Graph the arithmetic series showing the monthly sums of one year of Javier’s deposits.

Answer

Graph of Javier's deposits where the x-axis is the months of the year and the y-axis is the sum of deposits.

13.4-29 §13.4 source ↗

For the following exercises, use the geometric series ${\sum_{k = 1}^{\infty} (\frac{1}{2})}^{k} .$

What number does $S_{n}$ seem to be approaching in the graph? Find the sum to explain why this makes sense.

Answer

Sample answer: The graph of

S_{n}

seems to be approaching 1. This makes sense because

\sum_{k = 1}^{\infty} {(\frac{1}{2})}^{k}

is a defined infinite geometric series with

S =

\frac{\frac{1}{2}}{1 - (\frac{1}{2})} =

1.

13.4-31 §13.4 source ↗

For the following exercises, find the indicated sum.

$\sum_{n = 1}^{6} n (n - 2)$

Answer

13.4-33 §13.4 source ↗

For the following exercises, find the indicated sum.

$\sum_{k = 1}^{7} 2^{k}$

Answer

254

13.4-35 §13.4 source ↗

For the following exercises, use the formula for the sum of the first $n$ terms of an arithmetic series to find the sum.

$6 +$ $\frac{15}{2} +$ $9 +$ $\frac{21}{2} +$ $12 +$ $\frac{27}{2} +$ $15$

Answer

S_{7} = \frac{147}{2}

13.4-37 §13.4 source ↗

For the following exercises, use the formula for the sum of the first $n$ terms of an arithmetic series to find the sum.

$\sum_{k = 1}^{11} (\frac{k}{2} - \frac{1}{2})$

Answer

S_{11} = \frac{55}{2}

13.4-39 §13.4 source ↗

For the following exercises, use the formula for the sum of the first $n$ terms of a geometric series to find the partial sum.

$S_{7}$ for the series $0.4 -$ $2 +$ $10 -$ $50...$

Answer

S_{7} = 5208.4

13.4-41 §13.4 source ↗

For the following exercises, use the formula for the sum of the first $n$ terms of a geometric series to find the partial sum.

$\sum_{n = 1}^{10} - 2 \cdot {(\frac{1}{2})}^{n - 1}$

Answer

S_{10} =

-

\frac{1023}{256}

13.4-43 §13.4 source ↗

For the following exercises, find the sum of the infinite geometric series.

$-$ $1 -$ $\frac{1}{4} -$ $\frac{1}{16} -$ $\frac{1}{64} ...$

Answer

S = - \frac{4}{3}

13.4-45 §13.4 source ↗

For the following exercises, find the sum of the infinite geometric series.

$\sum_{n = 1}^{\infty} 4.6 \cdot {0.5}^{n - 1}$

Answer

S = 9.2

13.4-47 §13.4 source ↗

For the following exercises, determine the value of the annuity for the indicated monthly deposit amount, the number of deposits, and the interest rate.

Deposit amount: $$ 150;$ total deposits: $24;$ interest rate: $3 %,$ compounded monthly

Answer

$3,705.42

13.4-49 §13.4 source ↗

For the following exercises, determine the value of the annuity for the indicated monthly deposit amount, the number of deposits, and the interest rate.

Deposit amount: $$ 100;$ total deposits: $120;$ interest rate: $10 %,$ compounded semi-annually

Answer

$695,823.97

13.4-51 §13.4 source ↗

For the following exercises, determine the value of the annuity for the indicated monthly deposit amount, the number of deposits, and the interest rate.

Write an explicit formula for $a_{k}$ such that $\sum_{k = 0}^{6} a_{k} = 189.$ Assume this is an arithmetic series.

Answer

a_{k} = 30 - k

13.4-53 §13.4 source ↗

For the following exercises, determine the value of the annuity for the indicated monthly deposit amount, the number of deposits, and the interest rate.

How many terms must be added before the series $-$ $1 -$ $3 -$ $5 -$ $7 ....$ has a sum less than $- 75 ?$

Answer

9 terms

13.4-55 §13.4 source ↗

For the following exercises, determine the value of the annuity for the indicated monthly deposit amount, the number of deposits, and the interest rate.

The sum of an infinite geometric series is five times the value of the first term. What is the common ratio of the series?

Answer

r = \frac{4}{5}

13.4-57 §13.4 source ↗

For the following exercises, determine the value of the annuity for the indicated monthly deposit amount, the number of deposits, and the interest rate.

Karl has two years to save $$ 10, 000$ to buy a used car when he graduates. To the nearest dollar, what would his monthly deposits need to be if he invests in an account offering a 4.2% annual interest rate that compounds monthly?

Answer

$400 per month

13.4-59 §13.4 source ↗

For the following exercises, determine the value of the annuity for the indicated monthly deposit amount, the number of deposits, and the interest rate.

A boulder rolled down a mountain, traveling 6 feet in the first second. Each successive second, its distance increased by 8 feet. How far did the boulder travel after 10 seconds?

Answer

420 feet

13.4-61 §13.4 source ↗

For the following exercises, determine the value of the annuity for the indicated monthly deposit amount, the number of deposits, and the interest rate.

A pendulum travels a distance of 3 feet on its first swing. On each successive swing, it travels $\frac{3}{4}$ the distance of the previous swing. What is the total distance traveled by the pendulum when it stops swinging?

Answer

12 feet

Try It #1 §13.5 source ↗

A student is shopping for a new computer. He is deciding among 3 desktop computers and 4 laptop computers. What is the total number of computer options?

Answer

Try It #2 §13.5 source ↗

A restaurant offers a breakfast special that includes a breakfast sandwich, a side dish, and a beverage. There are 3 types of breakfast sandwiches, 4 side dish options, and 5 beverage choices. Find the total number of possible breakfast specials.

Answer

There are 60 possible breakfast specials.

Try It #3 §13.5 source ↗

A family of five is having portraits taken. Use the Multiplication Principle to find the following.

How many ways can the family line up for the portrait?

Answer

120

Try It #4 §13.5 source ↗

A family of five is having portraits taken. Use the Multiplication Principle to find the following.

How many ways can the photographer line up 3 family members?

Answer

Try It #5 §13.5 source ↗

A family of five is having portraits taken. Use the Multiplication Principle to find the following.

How many ways can the family line up for the portrait if the parents are required to stand on each end?

Answer

Try It #6 §13.5 source ↗

A play has a cast of 7 actors preparing to make their curtain call. Use the permutation formula to find the following.

How many ways can the 7 actors line up?

Answer

P (7, 7) =

5,

040

Try It #7 §13.5 source ↗

A play has a cast of 7 actors preparing to make their curtain call. Use the permutation formula to find the following.

How many ways can 5 of the 7 actors be chosen to line up?

Answer

P (7, 5) =

2,

520

Try It #8 §13.5 source ↗

An ice cream shop offers 10 flavors of ice cream. How many ways are there to choose 3 flavors for a banana split?

Answer

C (10, 3) = 120

Try It #9 §13.5 source ↗

A sundae bar at a wedding has 6 toppings to choose from. Any number of toppings can be chosen. How many different sundaes are possible?

Answer

64 sundaes

Try It #10 §13.5 source ↗

Find the number of rearrangements of the letters in the word CARRIER.

Answer

840

13.5-1 §13.5 source ↗

For the following exercises, assume that there are $n$ ways an event $A$ can happen, $m$ ways an event $B$ can happen, and that $A and B$ are non-overlapping.

Use the Addition Principle of counting to explain how many ways event $A or B$ can occur.

Answer

There are

m + n

ways for either event

A

or event

B

to occur.

13.5-3 §13.5 source ↗

Answer the following questions.

When given two separate events, how do we know whether to apply the Addition Principle or the Multiplication Principle when calculating possible outcomes? What conjunctions may help to determine which operations to use?

Answer

The addition principle is applied when determining the total possible of outcomes of either event occurring. The multiplication principle is applied when determining the total possible outcomes of both events occurring. The word “or” usually implies an addition problem. The word “and” usually implies a multiplication problem.

13.5-5 §13.5 source ↗

Answer the following questions.

What is the term for the arrangement that selects $r$ objects from a set of $n$ objects when the order of the $r$ objects is not important? What is the formula for calculating the number of possible outcomes for this type of arrangement?

Answer

A combination;

C (n, r) =

\frac{n!}{(n - r)! r!}

13.5-7 §13.5 source ↗

For the following exercises, determine whether to use the Addition Principle or the Multiplication Principle. Then perform the calculations.

Let the set $B =$ ${- 23, - 16, - 7, - 2, 20, 36, 48, 72} .$ How many ways are there to choose a positive or an odd number from $A ?$

Answer

4 + 2 = 6

13.5-9 §13.5 source ↗

For the following exercises, determine whether to use the Addition Principle or the Multiplication Principle. Then perform the calculations.

How many ways are there to pick a paint color from 5 shades of green, 4 shades of blue, or 7 shades of yellow?

Answer

5 + 4 + 7 = 16

13.5-11 §13.5 source ↗

For the following exercises, determine whether to use the Addition Principle or the Multiplication Principle. Then perform the calculations.

How many outcomes are possible from tossing a coin and rolling a 6-sided die?

Answer

2 \times 6 = 12

13.5-13 §13.5 source ↗

For the following exercises, determine whether to use the Addition Principle or the Multiplication Principle. Then perform the calculations.

How many ways are there to construct a string of 3 digits if numbers can be repeated?

Answer

10^{3} = 1000

13.5-15 §13.5 source ↗

For the following exercises, compute the value of the expression.

$P (5, 2)$

Answer

P (5, 2) = 20

13.5-17 §13.5 source ↗

For the following exercises, compute the value of the expression.

$P (3, 3)$

Answer

P (3, 3) = 6

13.5-19 §13.5 source ↗

For the following exercises, compute the value of the expression.

$P (11, 5)$

Answer

P (11, 5) =

55,

440

13.5-21 §13.5 source ↗

For the following exercises, compute the value of the expression.

$C (12, 4)$

Answer

C (12, 4) = 495

13.5-23 §13.5 source ↗

For the following exercises, compute the value of the expression.

$C (7, 6)$

Answer

C (7, 6) = 7

13.5-25 §13.5 source ↗

For the following exercises, find the number of subsets in each given set.

${1, 2, 3, 4, 5, 6, 7, 8, 9, 10}$

Answer

2^{10} = 1024

13.5-27 §13.5 source ↗

For the following exercises, find the number of subsets in each given set.

A set containing 5 distinct numbers, 4 distinct letters, and 3 distinct symbols

Answer

2^{12} = 4096

13.5-29 §13.5 source ↗

For the following exercises, find the number of subsets in each given set.

The set of two-digit numbers between 1 and 100 containing the digit 0

Answer

2^{9} = 512

13.5-31 §13.5 source ↗

For the following exercises, find the distinct number of arrangements.

The letters in the word “academia”

Answer

\frac{8!}{3!} = 6720

13.5-33 §13.5 source ↗

For the following exercises, find the distinct number of arrangements.

The symbols in the string #,#,#,@,@,$,$,$,%,%,%,%

Answer

\frac{12!}{3! 2! 3! 4!}

13.5-35 §13.5 source ↗

For the following exercises, find the distinct number of arrangements.

The set, $S$ consists of $900,000,000$ whole numbers, each being the same number of digits long. How many digits long is a number from $S ?$ (Hint: use the fact that a whole number cannot start with the digit 0.)

Answer

13.5-37 §13.5 source ↗

For the following exercises, find the distinct number of arrangements.

Can $C (n, r)$ ever equal $P (n, r) ?$ Explain.

Answer

Yes, for the trivial cases

r = 0

and

r = 1.

r = 0,

then

C (n, r) =

P (n, r) =

1 .

r = 1,

then

r = 1,

C (n, r) =

P (n, r) =

n .

13.5-39 §13.5 source ↗

For the following exercises, find the distinct number of arrangements.

How many arrangements can be made from the letters of the word “mountains” if all the vowels must form a string?

Answer

\frac{6!}{2!} \times 4! =

8640

13.5-41 §13.5 source ↗

For the following exercises, find the distinct number of arrangements.

A cell phone company offers 6 different voice packages and 8 different data packages. Of those, 3 packages include both voice and data. How many ways are there to choose either voice or data, but not both?

Answer

6 - 3 + 8 - 3 = 8

13.5-43 §13.5 source ↗

For the following exercises, find the distinct number of arrangements.

A wholesale T-shirt company offers sizes small, medium, large, and extra-large in organic or non-organic cotton and colors white, black, gray, blue, and red. How many different T-shirts are there to choose from?

Answer

4 \times 2 \times 5 = 40

13.5-45 §13.5 source ↗

For the following exercises, find the distinct number of arrangements.

An art store has 4 brands of paint pens in 12 different colors and 3 types of ink. How many paint pens are there to choose from?

Answer

4 \times 12 \times 3 = 144

13.5-47 §13.5 source ↗

For the following exercises, find the distinct number of arrangements.

How many ways can a baseball coach arrange the order of 9 batters if there are 15 players on the team?

Answer

P (15, 9) =

1,

816,

214,

400

13.5-49 §13.5 source ↗

For the following exercises, find the distinct number of arrangements.

A motorcycle shop has 10 choppers, 6 bobbers, and 5 café racers—different types of vintage motorcycles. How many ways can the shop choose 3 choppers, 5 bobbers, and 2 café racers for a weekend showcase?

Answer

C (10, 3) \times C (6, 5) \times C (5, 2) =

7,

200

13.5-51 §13.5 source ↗

For the following exercises, find the distinct number of arrangements.

Just-For-Kicks Sneaker Company offers an online customizing service. How many ways are there to design a custom pair of Just-For-Kicks sneakers if a customer can choose from a basic shoe up to 11 customizable options?

Answer

2^{11} = 2048

13.5-53 §13.5 source ↗

For the following exercises, find the distinct number of arrangements.

Suni bought 20 plants to arrange along the border of her garden. How many distinct arrangements can she make if the plants are comprised of 6 tulips, 6 roses, and 8 daisies?

Answer

\frac{20!}{6! 6! 8!} =

116,

396,

280

Try It #1 §13.6 source ↗

Find each binomial coefficient.

ⓐ $(\begin{matrix} 7 \\ 3 \end{matrix})$

ⓑ $(\begin{matrix} 11 \\ 4 \end{matrix})$

Answer

ⓐ35

ⓑ330

Try It #2 §13.6 source ↗

Write in expanded form.

ⓐ ${(x - y)}^{5}$

ⓑ ${(2 x + 5 y)}^{3}$

Answer

ⓐ $x^{5} -$ $5 x^{4} y +$ $10 x^{3} y^{2} -$ $10 x^{2} y^{3} +$ $5 x y^{4} -$ $y^{5}$

ⓑ $8 x^{3} +$ $60 x^{2} y +$ $150 x y^{2} +$ $125 y^{3}$

Try It #3 §13.6 source ↗

Find the sixth term of

{(3 x - y)}^{9}

without fully expanding the binomial.

Answer

- 10, 206 x^{4} y^{5}

13.6-1 §13.6 source ↗

What is a binomial coefficient, and how it is calculated?

Answer

A binomial coefficient is an alternative way of denoting the combination

C (n, r).

It is defined as

(\begin{matrix} n \\ r \end{matrix}) = C (n, r) = \frac{n!}{r! (n - r)!} .

13.6-3 §13.6 source ↗

What is the Binomial Theorem and what is its use?

Answer

The Binomial Theorem is defined as

{(x + y)}^{n} = \sum_{k = 0}^{n} (\begin{matrix} n \\ k \end{matrix}) x^{n - k} y^{k}

and can be used to expand any binomial.

13.6-5 §13.6 source ↗

For the following exercises, evaluate the binomial coefficient.

$(\begin{matrix} 6 \\ 2 \end{matrix})$

Answer

13.6-7 §13.6 source ↗

For the following exercises, evaluate the binomial coefficient.

$(\begin{matrix} 7 \\ 4 \end{matrix})$

Answer

13.6-9 §13.6 source ↗

For the following exercises, evaluate the binomial coefficient.

$(\begin{matrix} 10 \\ 9 \end{matrix})$

Answer

13.6-11 §13.6 source ↗

For the following exercises, evaluate the binomial coefficient.

$(\begin{matrix} 17 \\ 6 \end{matrix})$

Answer

12,376

13.6-13 §13.6 source ↗

For the following exercises, use the Binomial Theorem to expand each binomial.

${(4 a - b)}^{3}$

Answer

64 a^{3} -

48 a^{2} b +

12 a b^{2} -

b^{3}

13.6-15 §13.6 source ↗

For the following exercises, use the Binomial Theorem to expand each binomial.

${(3 a + 2 b)}^{3}$

Answer

27 a^{3} +

54 a^{2} b +

36 a b^{2} +

8 b^{3}

13.6-17 §13.6 source ↗

For the following exercises, use the Binomial Theorem to expand each binomial.

${(4 x + 2 y)}^{5}$

Answer

1024 x^{5} +

2560 x^{4} y +

2560 x^{3} y^{2} +

1280 x^{2} y^{3} +

320 x y^{4} +

32 y^{5}

13.6-19 §13.6 source ↗

For the following exercises, use the Binomial Theorem to expand each binomial.

${(4 x - 3 y)}^{5}$

Answer

1024 x^{5} -

3840 x^{4} y +

5760 x^{3} y^{2} -

4320 x^{2} y^{3} +

1620 x y^{4} -

243 y^{5}

13.6-21 §13.6 source ↗

For the following exercises, use the Binomial Theorem to expand each binomial.

${(x^{- 1} + 2 y^{- 1})}^{4}$

Answer

\frac{1}{x^{4}} +

\frac{8}{x^{3} y} +

\frac{24}{x^{2} y^{2}} +

\frac{32}{x y^{3}} +

\frac{16}{y^{4}}

13.6-23 §13.6 source ↗

For the following exercises, use the Binomial Theorem to write the first three terms of each binomial.

${(a + b)}^{17}$

Answer

a^{17} +

17 a^{16} b +

136 a^{15} b^{2}

13.6-25 §13.6 source ↗

For the following exercises, use the Binomial Theorem to write the first three terms of each binomial.

${(a - 2 b)}^{15}$

Answer

a^{15} -

30 a^{14} b +

420 a^{13} b^{2}

13.6-27 §13.6 source ↗

For the following exercises, use the Binomial Theorem to write the first three terms of each binomial.

${(3 a + b)}^{20}$

Answer

3,

486,

784,

401 a^{20} +

23,

245,

229,

340 a^{19} b +

73,

609,

892,

910 a^{18} b^{2}

13.6-29 §13.6 source ↗

For the following exercises, use the Binomial Theorem to write the first three terms of each binomial.

${(x^{3} - \sqrt{y})}^{8}$

Answer

x^{24} -

8 x^{21} \sqrt{y} +

28 x^{18} y

13.6-31 §13.6 source ↗

For the following exercises, find the indicated term of each binomial without fully expanding the binomial.

The fourth term of ${(3 x - 2 y)}^{5}$

Answer

- 720 x^{2} y^{3}

13.6-33 §13.6 source ↗

For the following exercises, find the indicated term of each binomial without fully expanding the binomial.

The eighth term of ${(7 + 5 y)}^{14}$

Answer

220,

812,

466,

875,

000 y^{7}

13.6-35 §13.6 source ↗

For the following exercises, find the indicated term of each binomial without fully expanding the binomial.

The fifth term of ${(x - y)}^{7}$

Answer

35 x^{3} y^{4}

13.6-37 §13.6 source ↗

For the following exercises, find the indicated term of each binomial without fully expanding the binomial.

The ninth term of ${(a - 3 b^{2})}^{11}$

Answer

1,

082,

565 a^{3} b^{16}

13.6-39 §13.6 source ↗

For the following exercises, find the indicated term of each binomial without fully expanding the binomial.

The eighth term of ${(\frac{y}{2} + \frac{2}{x})}^{9}$

Answer

\frac{1152 y^{2}}{x^{7}}

13.6-41 §13.6 source ↗

For the following exercises, use the Binomial Theorem to expand the binomial $f (x) =$ ${(x + 3)}^{4} .$ Then find and graph each indicated sum on one set of axes.

Find and graph $f_{2} (x),$ such that $f_{2} (x)$ is the sum of the first two terms of the expansion.

Answer

$f_{2} (x) =$ $x^{4} +$ $12 x^{3}$

13.6-43 §13.6 source ↗

For the following exercises, use the Binomial Theorem to expand the binomial $f (x) =$ ${(x + 3)}^{4} .$ Then find and graph each indicated sum on one set of axes.

Find and graph $f_{4} (x),$ such that $f_{4} (x)$ is the sum of the first four terms of the expansion.

Answer

$f_{4} (x) =$ $x^{4} +$ $12 x^{3} +$ $54 x^{2} +$ $108 x$

13.6-45 §13.6 source ↗

For the following exercises, use the Binomial Theorem to expand the binomial $f (x) =$ ${(x + 3)}^{4} .$ Then find and graph each indicated sum on one set of axes.

In the expansion of ${(5 x + 3 y)}^{n},$ each term has the form $(\begin{matrix} n \\ k \end{matrix}) a^{n - k} b^{k}$ , where $k$ successively takes on the value $0,$ $1,$ $2,$ $...,$ $n .$ If $(\begin{matrix} n \\ k \end{matrix}) = (\begin{matrix} 7 \\ 2 \end{matrix}),$ what is the corresponding term?

Answer

590, 625 x^{5} y^{2}

13.6-47 §13.6 source ↗

For the following exercises, use the Binomial Theorem to expand the binomial $f (x) =$ ${(x + 3)}^{4} .$ Then find and graph each indicated sum on one set of axes.

Consider the expansion of ${(x + b)}^{40} .$ What is the exponent of $b$ in the $k th$ term?

Answer

k - 1

13.6-49 §13.6 source ↗

For the following exercises, use the Binomial Theorem to expand the binomial $f (x) =$ ${(x + 3)}^{4} .$ Then find and graph each indicated sum on one set of axes.

Which expression cannot be expanded using the Binomial Theorem? Explain.

• $(x^{2} - 2 x + 1)$

• ${(\sqrt{a} + 4 \sqrt{a} - 5)}^{8}$

• ${(x^{3} + 2 y^{2} - z)}^{5}$

• ${(3 x^{2} - \sqrt{2 y^{3}})}^{12}$

Answer

The expression

{(x^{3} + 2 y^{2} - z)}^{5}

cannot be expanded using the Binomial Theorem because it cannot be rewritten as a binomial.

Try It #1 §13.7 source ↗

Construct a probability model for tossing a fair coin.

Answer

Outcome	Probability
Heads	$\frac{1}{2}$
Tails	$\frac{1}{2}$

Try It #2 §13.7 source ↗

A number cube is rolled. Find the probability of rolling a number greater than 2.

Answer

\frac{2}{3}

Try It #3 §13.7 source ↗

A card is drawn from a standard deck. Find the probability of drawing a red card or an ace.

Answer

\frac{7}{13}

Try It #4 §13.7 source ↗

A card is drawn from a standard deck. Find the probability of drawing an ace or a king.

Answer

\frac{2}{13}

Try It #5 §13.7 source ↗

Two number cubes are rolled. Use the Complement Rule to find the probability that the sum is less than 10.

Answer

\frac{5}{6}

Try It #6 §13.7 source ↗

A child randomly selects 3 gumballs from a container holding 4 purple gumballs, 8 yellow gumballs, and 2 green gumballs.

ⓐFind the probability that all 3 gumballs selected are purple.

ⓑFind the probability that no yellow gumballs are selected.

Answer

\begin{array}{l} a . \frac{1}{91}; & b . \frac{5}{91}; & c . \frac{86}{91} \end{array}

13.7-1 §13.7 source ↗

What term is used to express the likelihood of an event occurring? Are there restrictions on its values? If so, what are they? If not, explain.

Answer

probability; The probability of an event is restricted to values between

0

and

1,

inclusive of

0

and

1.

13.7-3 §13.7 source ↗

What is an experiment?

Answer

An experiment is an activity with an observable result.

13.7-5 §13.7 source ↗

The union of two sets is defined as a set of elements that are present in at least one of the sets. How is this similar to the definition used for the union of two events from a probability model? How is it different?

Answer

The probability of the union of two events occurring is a number that describes the likelihood that at least one of the events from a probability model occurs. In both a union of sets

A and B

and a union of events

A and B,

the union includes either

A or B

or both. The difference is that a union of sets results in another set, while the union of events is a probability, so it is always a numerical value between

0

and

1.

13.7-7 §13.7 source ↗

For the following exercises, use the spinner shown in Figure 3 to find the probabilities indicated.

A pie chart with eight pieces with one A colored blue, one B colored purple, once C colored orange, one D colored blue, one E colored red, one F colored green, one I colored green, and one O colored yellow.

Figure 3

Landing on a vowel

Answer

\frac{1}{2} .

13.7-9 §13.7 source ↗

For the following exercises, use the spinner shown in Figure 3 to find the probabilities indicated.

Figure 3

Landing on purple or a vowel

Answer

\frac{5}{8} .

13.7-11 §13.7 source ↗

For the following exercises, use the spinner shown in Figure 3 to find the probabilities indicated.

Figure 3

Landing on green or blue

Answer

\frac{1}{2} .

13.7-13 §13.7 source ↗

For the following exercises, use the spinner shown in Figure 3 to find the probabilities indicated.

Figure 3

Not landing on yellow or a consonant

Answer

\frac{3}{8} .

13.7-15 §13.7 source ↗

For the following exercises, two coins are tossed.

Find the probability of tossing two heads.

Answer

\frac{1}{4} .

13.7-17 §13.7 source ↗

For the following exercises, two coins are tossed.

Find the probability of tossing at least one tail.

Answer

\frac{3}{4} .

13.7-19 §13.7 source ↗

For the following exercises, four coins are tossed.

Find the probability of tossing exactly two heads.

Answer

\frac{3}{8} .

13.7-21 §13.7 source ↗

For the following exercises, four coins are tossed.

Find the probability of tossing four heads or four tails.

Answer

\frac{1}{8} .

13.7-23 §13.7 source ↗

For the following exercises, four coins are tossed.

Find the probability of tossing not all tails.

Answer

\frac{15}{16} .

13.7-25 §13.7 source ↗

For the following exercises, four coins are tossed.

Find the probability of tossing either two heads or three heads.

Answer

\frac{5}{8} .

13.7-27 §13.7 source ↗

For the following exercises, one card is drawn from a standard deck of $52$ cards. Find the probability of drawing the following:

A two

Answer

\frac{1}{13} .

13.7-29 §13.7 source ↗

For the following exercises, one card is drawn from a standard deck of $52$ cards. Find the probability of drawing the following:

Red six

Answer

\frac{1}{26} .

13.7-31 §13.7 source ↗

For the following exercises, one card is drawn from a standard deck of $52$ cards. Find the probability of drawing the following:

A non-ace

Answer

\frac{12}{13} .

13.7-33 §13.7 source ↗

For the following exercises, two dice are rolled, and the results are summed.

Construct a table showing the sample space of outcomes and sums.

Answer

	1	2	3	4	5	6
1	(1,1) 2	(1,2) 3	(1,3) 4	(1,4) 5	(1,5) 6	(1,6) 7
2	(2,1) 3	(2,2) 4	(2,3) 5	(2,4) 6	(2,5) 7	(2,6) 8
3	(3,1) 4	(3,2) 5	(3,3) 6	(3,4) 7	(3,5) 8	(3,6) 9
4	(4,1) 5	(4,2) 6	(4,3) 7	(4,4) 8	(4,5) 9	(4,6) 10
5	(5,1) 6	(5,2) 7	(5,3) 8	(5,4) 9	(5,5) 10	(5,6) 11
6	(6,1) 7	(6,2) 8	(6,3) 9	(6,4) 10	(6,5) 11	(6,6) 12

13.7-35 §13.7 source ↗

For the following exercises, two dice are rolled, and the results are summed.

Find the probability of rolling at least one four or a sum of $8.$

Answer

\frac{5}{12} .

13.7-37 §13.7 source ↗

For the following exercises, two dice are rolled, and the results are summed.

Find the probability of rolling a sum greater than or equal to $15.$

Answer

0.

13.7-39 §13.7 source ↗

For the following exercises, two dice are rolled, and the results are summed.

Find the probability of rolling a sum less than $6$ or greater than $9.$

Answer

\frac{4}{9} .

13.7-41 §13.7 source ↗

For the following exercises, two dice are rolled, and the results are summed.

Find the probability of rolling a sum of $5$ or $6.$

Answer

\frac{1}{4} .

13.7-43 §13.7 source ↗

For the following exercises, a coin is tossed, and a card is pulled from a standard deck. Find the probability of the following:

A head on the coin or a club

Answer

\frac{5}{8}

13.7-45 §13.7 source ↗

For the following exercises, a coin is tossed, and a card is pulled from a standard deck. Find the probability of the following:

A head on the coin or a face card

Answer

\frac{8}{13}

13.7-47 §13.7 source ↗

For the following exercises, use this scenario: a bag of M&Ms contains $12$ blue, $6$ brown, $10$ orange, $8$ yellow, $8$ red, and $4$ green M&Ms. Reaching into the bag, a person grabs 5 M&Ms.

What is the probability of getting all blue M&Ms?

Answer

\frac{C (12, 5)}{C (48, 5)} =

\frac{1}{2162}

13.7-49 §13.7 source ↗

For the following exercises, use this scenario: a bag of M&Ms contains $12$ blue, $6$ brown, $10$ orange, $8$ yellow, $8$ red, and $4$ green M&Ms. Reaching into the bag, a person grabs 5 M&Ms.

What is the probability of getting $3$ blue M&Ms?

Answer

\frac{C (12, 3) C (36, 2)}{C (48, 5)} =

\frac{175}{2162}

13.7-51 §13.7 source ↗

Use the following scenario for the exercises that follow: In the game of Keno, a player starts by selecting $20$ numbers from the numbers $1$ to $80.$ After the player makes his selections, $20$ winning numbers are randomly selected from numbers $1$ to $80.$ A win occurs if the player has correctly selected $3, 4,$ or $5$ of the $20$ winning numbers. (Round all answers to the nearest hundredth of a percent.)

What is the percent chance that a player selects exactly 3 winning numbers?

Answer

\frac{C (20, 3) C (60, 17)}{C (80, 20)} \approx

12.49 %

13.7-53 §13.7 source ↗

What is the percent chance that a player selects all 5 winning numbers?

Answer

\frac{C (20, 5) C (60, 15)}{C (80, 20)} \approx

23.33 %

13.7-55 §13.7 source ↗

How much less is a player’s chance of selecting 3 winning numbers than the chance of selecting either 4 or 5 winning numbers?

Answer

20.50 +

23.33 -

12.49 =

31.34 %

13.7-57 §13.7 source ↗

Use this data for the exercises that follow: In 2013, there were roughly 317 million citizens in the United States, and about 40 million were elderly (aged 65 and over).²

If you meet five U.S. citizens, what is the percent chance that exactly one is elderly? (Round to the nearest tenth of a percent.)

Answer

\frac{C (40000000, 1) C (277000000, 4)}{C (317000000, 5)} =

36.78 %

13.7-59 §13.7 source ↗

Use this data for the exercises that follow: In 2013, there were roughly 317 million citizens in the United States, and about 40 million were elderly (aged 65 and over).²

If you meet five U.S. citizens, what is the percent chance that four are elderly? (Round to the nearest thousandth of a percent.)

Answer

\frac{C (40000000, 4) C (277000000, 1)}{C (317000000, 5)} =

0.11 %

13.R-1 §13.R source ↗

Write the first four terms of the sequence defined by the recursive formula

a_{1} =

2,

a_{n} =

a_{n - 1} +

n .

Answer

2, 4, 7, 11

13.R-3 §13.R source ↗

Write the first four terms of the sequence defined by the explicit formula

a_{n} = 10^{n} + 3.

Answer

13,

103,

1003,

10003

13.R-5 §13.R source ↗

Is the sequence

\frac{4}{7},

\frac{47}{21},

\frac{82}{21},

\frac{39}{7},

...

arithmetic? If so, find the common difference.

Answer

The sequence is arithmetic. The common difference is

d = \frac{5}{3} .

13.R-7 §13.R source ↗

An arithmetic sequence has the first term

a_{1} = 18

and common difference

d = - 8.

What are the first five terms?

Answer

18,

10,

2,

-

6,

-

14

13.R-9 §13.R source ↗

Write a recursive formula for the arithmetic sequence

-

20, -

10,

0, 10,…

Answer

a_{1} =

-

20,

a_{n} =

a_{n - 1} +

10

13.R-11 §13.R source ↗

Write an explicit formula for the arithmetic sequence

\frac{7}{8},

\frac{29}{24},

\frac{37}{24},

\frac{15}{8},

\dots

Answer

a_{n} = \frac{1}{3} n + \frac{13}{24}

13.R-13 §13.R source ↗

Find the common ratio for the geometric sequence

2.5,

5,

10,

20,

\dots

Answer

r = 2

13.R-15 §13.R source ↗

A geometric sequence has terms

a_{7} = 16, 384

and

a_{9} = 262, 144

. What are the first five terms?

Answer

4, 16, 64, 256, 1024

13.R-17 §13.R source ↗

What are the first five terms of the geometric sequence

a_{1} =

3,

a_{n} =

4 \cdot a_{n - 1} ?

Answer

3,

12,

48,

192,

768

13.R-19 §13.R source ↗

Write an explicit formula for the geometric sequence

-

\frac{1}{5},

-

\frac{1}{15},

-

\frac{1}{45},

-

\frac{1}{135},

\dots

Answer

a_{n} =

-

\frac{1}{5} \cdot {(\frac{1}{3})}^{n - 1}

13.R-21 §13.R source ↗

Use summation notation to write the sum of terms

\frac{1}{2} m + 5

from

m = 0

m = 5.

Answer

\sum_{m = 0}^{5} (\frac{1}{2} m + 5) .

13.R-23 §13.R source ↗

Use the formula for the sum of the first

n

terms of an arithmetic series to find the sum of the first eleven terms of the arithmetic series 2.5, 4, 5.5, … .

Answer

S_{11} = 110

13.R-25 §13.R source ↗

Use the formula for the sum of the first n terms of a geometric series to find

S_{9}

for the series

12, 6, 3, \frac{3}{2}, \dots

Answer

S_{9} \approx 23.95

13.R-27 §13.R source ↗

Find the sum of the infinite geometric series

\sum_{k = 1}^{\infty} 45 \cdot {(- \frac{1}{3})}^{k - 1} .

Answer

S = \frac{135}{4}

13.R-29 §13.R source ↗

Alejandro deposits $80 of his monthly earnings into an annuity that earns 6.25% annual interest, compounded monthly. How much money will he have saved after 5 years?

Answer

$5,617.61

13.R-31 §13.R source ↗

How many ways are there to choose a number from the set

{- 10, - 6, 4, 10, 12, 18, 24, 32}

that is divisible by either

4

6 ?

Answer

13.R-33 §13.R source ↗

How many ways are there to construct a 4-digit code if numbers can be repeated?

Answer

10^{4} = 10, 000

13.R-35 §13.R source ↗

Calculate

P (18, 4) .

Answer

P (18, 4) =

73, 440

13.R-37 §13.R source ↗

Calculate

C (15, 6) .

Answer

C (15, 6) =

5005

13.R-39 §13.R source ↗

How many subsets does the set

{1, 3, 5, \dots, 99}

have?

Answer

2^{50} =

1.13 \times 10^{15}

13.R-41 §13.R source ↗

How many distinct ways can the word DEADWOOD be arranged?

Answer

\frac{8!}{3! 2!} = 3360

13.R-43 §13.R source ↗

Evaluate the binomial coefficient

(\begin{matrix} 23 \\ 8 \end{matrix}) .

Answer

490, 314

13.R-45 §13.R source ↗

Use the Binomial Theorem to write the first three terms of

{(2 a + b)}^{17} .

Answer

131, 072 a^{17} + 1, 114, 112 a^{16} b + 4, 456, 448 a^{15} b^{2}

13.R-47 §13.R source ↗

For the following exercises, assume two die are rolled.

Construct a table showing the sample space.

Answer

	1	2	3	4	5	6
1	1,1	1,2	1,3	1,4	1,5	1,6
2	2,1	2,2	2,3	2,4	2,5	2,6
3	3,1	3,2	3,3	3,4	3,5	3,6
4	4,1	4,2	4,3	4,4	4,5	4,6
5	5,1	5,2	5,3	5,4	5,5	5,6
6	6,1	6,2	6,3	6,4	6,5	6,6

13.R-49 §13.R source ↗

For the following exercises, assume two die are rolled.

What is the probability of rolling a pair?

Answer

\frac{1}{6}

13.R-51 §13.R source ↗

For the following exercises, assume two die are rolled.

What is the probability that a roll doesn’t include a 2 or result in a pair?

Answer

\frac{5}{9}

13.R-53 §13.R source ↗

For the following exercises, assume two die are rolled.

What is the probability that a roll includes neither a 5 nor a 6?

Answer

\frac{4}{9}

13.R-55 §13.R source ↗

For the following exercises, use the following data: An elementary school survey found that 350 of the 500 students preferred soda to milk. Suppose 8 children from the school are attending a birthday party. (Show calculations and round to the nearest tenth of a percent.)

What is the percent chance that at least one of the children attending the party prefers milk?

Answer

1 -

\frac{C (350, 8)}{C (500, 8)} \approx

94.4 %

13.R-57 §13.R source ↗

What is the percent chance that exactly 3 of the children attending the party prefer milk?

Answer

\frac{C (150, 3) C (350, 5)}{C (500, 8)} \approx

25.6 %

13.T-1 §13.T source ↗

Write the first four terms of the sequence defined by the recursive formula

a =

- 14,

a_{n} =

\frac{2 + a_{n - 1}}{2} .

Answer

- 14, - 6, - 2, 0

13.T-3 §13.T source ↗

Is the sequence

0.3,

1.2,

2.1,

3,

\dots

arithmetic? If so find the common difference.

Answer

The sequence is arithmetic. The common difference is

d = 0.9.

13.T-5 §13.T source ↗

Write a recursive formula for the arithmetic sequence

-

2,

-

\frac{7}{2},

-

5,

-

\frac{13}{2},

\dots

and then find the 22^nd term.

Answer

a_{1} =

-

2,

a_{n} =

a_{n - 1} -

\frac{3}{2};

a_{22} =

-

\frac{67}{2}

13.T-7 §13.T source ↗

Is the sequence

-

2, -

1, -

\frac{1}{2}, -

\frac{1}{4}, \dots

geometric? If so find the common ratio. If not, explain why.

Answer

The sequence is geometric. The common ratio is

r = \frac{1}{2} .

13.T-9 §13.T source ↗

Write a recursive formula for the geometric sequence

1,

-

\frac{1}{2},

\frac{1}{4},

-

\frac{1}{8},

\dots

Answer

a_{1} =

1,

a_{n} =

-

\frac{1}{2} \cdot a_{n}_{- 1}

13.T-11 §13.T source ↗

Use summation notation to write the sum of terms

3 k^{2} - \frac{5}{6} k

from

k = - 3

k = 15.

Answer

\sum_{k = - 3}^{15} (3 k^{2} - \frac{5}{6} k)

13.T-13 §13.T source ↗

Use the formula for the sum of the first

n

terms of a geometric series to find

\sum_{k = 1}^{7} - 0.2 \cdot {(- 5)}^{k - 1} .

Answer

S_{7} = - 2604.2

13.T-15 §13.T source ↗

Ramla deposits $3,600 into a retirement fund each year. The fund earns 7.5% annual interest, compounded monthly. If she opened her account when she was 20 years old, how much will she have by the time she’s 55? How much of that amount was interest earned?

Answer

Total in account:

$ 634,

261.20;

Interest earned:

$ 508, 261.20

13.T-17 §13.T source ↗

A buyer of a new sedan can custom order the car by choosing from 5 different exterior colors, 3 different interior colors, 2 sound systems, 3 motor designs, and either manual or automatic transmission. How many choices does the buyer have?

Answer

5 \times 3 \times 2 \times 3 \times 2 =

180

13.T-19 §13.T source ↗

A music group needs to choose 3 songs to play at the annual Battle of the Bands. How many ways can they choose their set if they have 15 songs to pick from?

Answer

C (15, 3) = 455

13.T-21 §13.T source ↗

How many distinct ways can the word EVANESCENCE be arranged if the anagram must end with the letter E?

Answer

\frac{10!}{2! 3! 2!} =

151, 200

13.T-23 §13.T source ↗

Find the seventh term of

{(x^{2} - \frac{1}{2})}^{13}

without fully expanding the binomial.

Answer

\frac{429 x^{14}}{16}

13.T-25 §13.T source ↗

For the following exercises, use the spinner in Figure 1.

A circular spinner divided into seven distinct, numbered sections (1-7), each a different color, with a pointer indicating a value in the seventh section.

Figure 1

What is the probability of landing on an odd number?

Answer

\frac{4}{7}

13.T-27 §13.T source ↗

For the following exercises, use the spinner in Figure 1.

Figure 1

What is the probability of landing on blue or an odd number?

Answer

\frac{5}{7}

13.T-29 §13.T source ↗

For the following exercises, use the spinner in Figure 1.

Figure 1

A bowl of candy holds 16 peppermint, 14 butterscotch, and 10 strawberry flavored candies. Suppose a person grabs a handful of 7 candies. What is the percent chance that exactly 3 are butterscotch? (Show calculations and round to the nearest tenth of a percent.)

Answer

\frac{C (14, 3) C (26, 4)}{C (40, 7)} \approx

29.2 %

End of the book.

Problems and answers are adapted from Algebra and Trigonometry 2e by Jay Abramson, OpenStax, and are used under the CC BY-NC-SA 4.0 licence. Access for free at https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-introduction-to-prerequisites. Illustrations: Copyright Rice University, OpenStax, under CC BY-NC-SA 4.0 license. This adaptation is released under the same licence, is non-commercial, and is neither affiliated with nor endorsed by OpenStax or Rice University.

	1	2	3	4	5	6
1	1,1	1,2	1,3	1,4	1,5	1,6
2	2,1	2,2	2,3	2,4	2,5	2,6
3	3,1	3,2	3,3	3,4	3,5	3,6
4	4,1	4,2	4,3	4,4	4,5	4,6
5	5,1	5,2	5,3	5,4	5,5	5,6
6	6,1	6,2	6,3	6,4	6,5	6,6

	1	2	3	4	5	6
1	1,1	1,2	1,3	1,4	1,5	1,6
2	2,1	2,2	2,3	2,4	2,5	2,6
3	3,1	3,2	3,3	3,4	3,5	3,6
4	4,1	4,2	4,3	4,4	4,5	4,6
5	5,1	5,2	5,3	5,4	5,5	5,6
6	6,1	6,2	6,3	6,4	6,5	6,6

	1	2	3	4	5	6
1	1,1	1,2	1,3	1,4	1,5	1,6
2	2,1	2,2	2,3	2,4	2,5	2,6
3	3,1	3,2	3,3	3,4	3,5	3,6
4	4,1	4,2	4,3	4,4	4,5	4,6
5	5,1	5,2	5,3	5,4	5,5	5,6
6	6,1	6,2	6,3	6,4	6,5	6,6