Chapter 12: Sequences, Series and Binomial TheoremContent check — Intermediate Algebra, Chapter 12: Sequences, Series and Binomial Theorem

Evaluate $2n+3$ for the integers 1, 2, 3, and 4.
If you missed this problem, review Example 1.6.

Evaluate ${\left(\mathrm{-1}\right)}^n$ for the integers 1, 2, 3, and 4.
If you missed this problem, review Example 1.19.

If $f\left(n\right)=n^2+2,$ find $f\left(1\right)+$ $f\left(2\right)+$ $f\left(3\right).$
If you missed this problem, review Example 3.49.

Write the first five terms of the sequence whose general term is $a_n=3n-4.$

Write the first five terms of the sequence whose general term is $a_n=2n-5.$

Write the first five terms of the sequence whose general term is $a_n=3^n+4.$

Write the first five terms of the sequence whose general term is $a_n=2^n-5.$

Write the first five terms of the sequence whose general term is $a_n={\left(\mathrm{-1}\right)}^n{n}^2.$

Write the first five terms of the sequence whose general term is $a_n=$ ${\left(\mathrm{-1}\right)}^{n+1}{n}^3.$

Find a general term for the sequence whose first five terms are shown.

$3,$ $6,$ $9,$ $12,$ $15,$ $\text{…}$

Find a general term for the sequence whose first five terms are shown.

$5,$ $10,$ $15,$ $20,$ $25,$ $\text{…}$

Find a general term for the sequence whose first five terms are shown.

$\mathrm{-3},$ $9,$ $\mathrm{-27},$ $81,$ $\mathrm{-243}\text{,}\text{…}$

Find a general term for the sequence whose first five terms are shown

$1,$ $\mathrm{-4},$ $9,$ $\mathrm{-16},$ $25\text{,}\text{…}$

Find a general term for the sequence whose first five terms are shown.

$\frac{1}{2},$ $\frac{1}{4},$ $\frac{1}{8},$ $\frac{1}{16},$ $\frac{1}{32},$ $\text{…}$

Find a general term for the sequence whose first five terms are shown.

$\frac{1}{1},$ $\frac{1}{4},$ $\frac{1}{9},$ $\frac{1}{16},$ $\frac{1}{25},$ $\text{…}$

Write the first five terms of the sequence whose general term is $a_n=\frac{2}{n!}.$

Write the first five terms of the sequence whose general term is $a_n=\frac{3}{n!}.$

Write the first five terms of the sequence whose general term is $a_n=$ $\frac{\left(n-1\right)!}{\left(n+1\right)!}.$

Write the first five terms of the sequence whose general term is $a_n=$ $\frac{n!}{\left(n+1\right)!}.$

Expand the partial sum and find its value: ${\displaystyle \sum _{i=1}^{5}3i}.$

Expand the partial sum and find its value: ${\displaystyle \sum _{i=1}^{5}4i}.$

Expand the partial sum and find its value: ${\displaystyle \sum _{k=0}^3\frac{2}{k!}}.$

Expand the partial sum and find its value: ${\displaystyle \sum _{k=0}^3\frac{3}{k!}}.$

Write the sum using summation notation: $\frac{1}{2}+$ $\frac{1}{4}+$ $\frac{1}{8}+$ $\frac{1}{16}+$ $\frac{1}{32}.$

Write the sum using summation notation: $1+$ $\frac{1}{4}+$ $\frac{1}{9}+$ $\frac{1}{16}+$ $\frac{1}{25}.$

Write each sum using summation notation: $1-$ $4+$ $9-$ $16+$ $25.$

Write each sum using summation notation: $\mathrm{-2}+$ $4-$ $6+$ $8-$ $10.$

Write the First Few Terms of a Sequence

In the following exercises, write the first five terms of the sequence whose general term is given.

$a_n=2n-7$

Write the First Few Terms of a Sequence

In the following exercises, write the first five terms of the sequence whose general term is given.

$a_n=3n+1$

Write the First Few Terms of a Sequence

In the following exercises, write the first five terms of the sequence whose general term is given.

$a_n=2^n+3$

Write the First Few Terms of a Sequence

In the following exercises, write the first five terms of the sequence whose general term is given.

$a_n=3^n-2n$

Write the First Few Terms of a Sequence

In the following exercises, write the first five terms of the sequence whose general term is given.

$a_n=\frac{2^n}{n^2}$

Write the First Few Terms of a Sequence

In the following exercises, write the first five terms of the sequence whose general term is given.

$a_n=\frac{4n-2}{2^n}$

Write the First Few Terms of a Sequence

In the following exercises, write the first five terms of the sequence whose general term is given.

$a_n={\left(\mathrm{-1}\right)}^n·2n$

Write the First Few Terms of a Sequence

In the following exercises, write the first five terms of the sequence whose general term is given.

$a_n=$ ${\left(\mathrm{-1}\right)}^{n+1}{n}^2$

Write the First Few Terms of a Sequence

In the following exercises, write the first five terms of the sequence whose general term is given.

$a_n=$ $\frac{{\left(\mathrm{-1}\right)}^{n+1}}{n^2}$

Find a Formula for the General Term (nth Term) of a Sequence

In the following exercises, find a general term for the sequence whose first five terms are shown.

$8,$ $16,$ $24,$ $32,$ $40,$ $\text{…}$

Find a Formula for the General Term (nth Term) of a Sequence

In the following exercises, find a general term for the sequence whose first five terms are shown.

$6,$ $7,$ $8,$ $9,$ $10,$ $\text{…}$

Find a Formula for the General Term (nth Term) of a Sequence

In the following exercises, find a general term for the sequence whose first five terms are shown.

$e^3,$ $e^4,$ $e^5,$ $e^6,$ $e^7,$ $\text{…}$

Find a Formula for the General Term (nth Term) of a Sequence

In the following exercises, find a general term for the sequence whose first five terms are shown.

$\mathrm{-5},$ $10,$ $\mathrm{-15},$ $20,$ $\mathrm{-25},$ $\text{…}$

Find a Formula for the General Term (nth Term) of a Sequence

In the following exercises, find a general term for the sequence whose first five terms are shown.

$\mathrm{-1},$ $8,$ $\mathrm{-27},$ $64,$ $\mathrm{-125},$ $\text{…}$

Find a Formula for the General Term (nth Term) of a Sequence

In the following exercises, find a general term for the sequence whose first five terms are shown.

$\mathrm{-2},$ $4,$ $\mathrm{-6},$ $8,$ $\mathrm{-10},$ $\text{…}$

Find a Formula for the General Term (nth Term) of a Sequence

In the following exercises, find a general term for the sequence whose first five terms are shown.

$\frac{1}{4},$ $\frac{1}{16},$ $\frac{1}{64},$ $\frac{1}{256},$ $\frac{1}{\mathrm{1,024}},$ $\text{…}$

Find a Formula for the General Term (nth Term) of a Sequence

In the following exercises, find a general term for the sequence whose first five terms are shown.

$-$ $\frac{1}{2},$ $-$ $\frac{2}{3},$ $-$ $\frac{3}{4},$ $-$ $\frac{4}{5},$ $-$ $\frac{5}{6},$ $\text{…}$

Find a Formula for the General Term (nth Term) of a Sequence

In the following exercises, find a general term for the sequence whose first five terms are shown.

$-$ $\frac{5}{2},$ $-$ $\frac{5}{4},$ $-$ $\frac{5}{8},$ $-$ $\frac{5}{16},$ $-$ $\frac{5}{32},$ $\text{…}$

Use Factorial Notation

In the following exercises, using factorial notation, write the first five terms of the sequence whose general term is given.

$a_n=\frac{4}{n!}$

Use Factorial Notation

In the following exercises, using factorial notation, write the first five terms of the sequence whose general term is given.

$a_n=3n!$

Use Factorial Notation

In the following exercises, using factorial notation, write the first five terms of the sequence whose general term is given.

$a_n=\left(2n\right)!$

Use Factorial Notation

In the following exercises, using factorial notation, write the first five terms of the sequence whose general term is given.

$a_n=$ $\frac{\left(n-1\right)!}{\left(n\right)!}$

Use Factorial Notation

In the following exercises, using factorial notation, write the first five terms of the sequence whose general term is given.

$a_n=\frac{n!}{n^{\text{2}}}$

Use Factorial Notation

In the following exercises, using factorial notation, write the first five terms of the sequence whose general term is given.

$a_n=\frac{\left(n+1\right)!}{n^2}$

Find the Partial Sum

In the following exercises, expand the partial sum and find its value.

${\displaystyle \sum _{i=1}^5{i}^2}$

Find the Partial Sum

In the following exercises, expand the partial sum and find its value.

${\displaystyle \sum _{i=1}^6(2i}+3)$

Find the Partial Sum

In the following exercises, expand the partial sum and find its value.

${\displaystyle \sum _{i=1}^4{2}^i}$

Find the Partial Sum

In the following exercises, expand the partial sum and find its value.

${\displaystyle \sum _{k=0}^3\frac{4}{k!}}$

Find the Partial Sum

In the following exercises, expand the partial sum and find its value.

${\displaystyle \sum _{k=1}^{5}k\left(k+1\right)}$

Find the Partial Sum

In the following exercises, expand the partial sum and find its value.

${\displaystyle \sum _{n=1}^5\frac{n}{n+1}}$

Use Summation Notation to write a Sum

In the following exercises, write each sum using summation notation.

$\frac{1}{3}+$ $\frac{1}{9}+$ $\frac{1}{27}+$ $\frac{1}{81}+$ $\frac{1}{243}$

Use Summation Notation to write a Sum

In the following exercises, write each sum using summation notation.

$1+$ $\frac{1}{8}+$ $\frac{1}{27}+$ $\frac{1}{64}+$ $\frac{1}{125}$

Use Summation Notation to write a Sum

In the following exercises, write each sum using summation notation.

$2+$ $1+$ $\frac{2}{3}+$ $\frac{1}{2}+$ $\frac{2}{5}$

Use Summation Notation to write a Sum

In the following exercises, write each sum using summation notation.

$3-6+9-12+15$

Use Summation Notation to write a Sum

In the following exercises, write each sum using summation notation.

$\mathrm{-2}+$ $4-$ $6+$ $8-$ $10+$ $\dots +$ $20$

Use Summation Notation to write a Sum

In the following exercises, write each sum using summation notation.

$14+$ $16+$ $18+$ $20+$ $22+$ $24+$ $26$

Use Summation Notation to write a Sum

In the following exercises, write each sum using summation notation.

In your own words, explain how to write the terms of a sequence when you know the formula. Show an example to illustrate your explanation.

Use Summation Notation to write a Sum

In the following exercises, write each sum using summation notation.

In your own words, explain what is meant by $n!$ Show some examples to illustrate your explanation.

Evaluate $4n-1$ for the integers 1, 2, 3, and 4.
If you missed this problem, review Example 1.6.

Solve the system of equations: $$\{\begin{array}{l}x+y=7\\ 3x+4y=23\end{array}.$$
If you missed this problem, review Example 4.9.

If $f\left(n\right)=$ $\frac{n}{2}(3n+5),$ find $f\left(1\right)+f\left(20\right).$
If you missed this problem, review Example 3.49.

Determine if each sequence is arithmetic. If so, indicate the common difference.

ⓐ $9,$ $20,$ $31,$ $42,$ $53,$ $64\text{,}\text{…}$ ⓑ $12,$ $6,$ $0,$ $\mathrm{-6},$ $\mathrm{-12},$ $\mathrm{-18}\text{,}\text{…}$ ⓒ $7,$ $1,$ $10,$ $4,$ $13,$ $7\text{,}\text{…}$

Determine if each sequence is arithmetic. If so, indicate the common difference.

ⓐ $\mathrm{-4},$ $4,$ $2,$ $10,$ $8,$ $16\text{,}\text{…}$ ⓑ $\mathrm{-3},$ $\mathrm{-1},$ $1,$ $3,$ $5,$ $7\text{,}\text{…}$ ⓒ $7,$ $2,$ $\mathrm{-3},$ $\mathrm{-8},$ $\mathrm{-13},$ $\mathrm{-18}\text{,}\text{…}$

Write the first five terms of the sequence where the first term is 7 and the common difference is $d=\mathrm{-4}.$

Write the first five terms of the sequence where the first term is 11 and the common difference is $d=\mathrm{-8}.$

Find the twenty-seventh term of a sequence where the first term is 7 and the common difference is 9.

Find the eighteenth term of a sequence where the first term is 13 and the common difference is $\mathrm{-7}$.

Find the eleventh term of a sequence where the ninth term is 8 and the common difference is $\mathrm{-3}.$ Give the formula for the general term.

Find the nineteenth term of a sequence where the fifth term is 1 and the common difference is $\mathrm{-4}.$ Give the formula for the general term.

Find the first term and common difference of a sequence where the fourth term is 17 and the thirteenth term is 53. Give the formula for the general term.

Find the first term and common difference of a sequence where the third term is 2 and the twelfth term is $\mathrm{-25}.$ Give the formula for the general term.

Find the sum of the first 30 terms of the arithmetic sequence: 5, 9, 13, 17, 21, …

Find the sum of the first 30 terms of the arithmetic sequence: 7, 10, 13, 16, 19, …

Find the sum of the first 50 terms of the arithmetic sequence whose general term is $a_n=2n-5.$

Find the sum of the first 50 terms of the arithmetic sequence whose general term is $a_n=4n+3.$

Find the sum: ${\displaystyle \sum _{i=1}^{30}(6i-4}).$

Find the sum: ${\displaystyle \sum _{i=1}^{35}(5i-3}).$

Determine if a Sequence is Arithmetic

In the following exercises, determine if each sequence is arithmetic, and if so, indicate the common difference.

$4,$ $12,$ $20,$ $28,$ $36,$ $44\text{,}\text{…}$

Determine if a Sequence is Arithmetic

In the following exercises, determine if each sequence is arithmetic, and if so, indicate the common difference.

$\mathrm{-15},$ $\mathrm{-16},$ $3,$ $12,$ $21,$ $30\text{,}\text{…}$

Determine if a Sequence is Arithmetic

In the following exercises, determine if each sequence is arithmetic, and if so, indicate the common difference.

$8,$ $5,$ $2,$ $\mathrm{-1},$ $\mathrm{-4},$ $\mathrm{-7}\text{,}\text{…}$

In the following exercises, write the first five terms of each sequence with the given first term and common difference.

$a_1=11$ and $d=7$

In the following exercises, write the first five terms of each sequence with the given first term and common difference.

$a_1=\mathrm{-7}$ and $d=4$

In the following exercises, write the first five terms of each sequence with the given first term and common difference.

$a_1=14$ and $d=\mathrm{-9}$

Find the General Term (nth Term) of an Arithmetic Sequence

In the following exercises, find the term described using the information provided.

Find the twenty-first term of a sequence where the first term is three and the common difference is eight.

Find the General Term (nth Term) of an Arithmetic Sequence

In the following exercises, find the term described using the information provided.

Find the thirtieth term of a sequence where the first term is $\mathrm{-14}$ and the common difference is five.

Find the General Term (nth Term) of an Arithmetic Sequence

In the following exercises, find the term described using the information provided.

Find the sixteenth term of a sequence where the first term is 11 and the common difference is $\mathrm{-6}.$

Find the General Term (nth Term) of an Arithmetic Sequence

In the following exercises, find the term described using the information provided.

Find the twentieth term of a sequence where the fifth term is $\mathrm{-4}$ and the common difference is $\mathrm{-2}.$ Give the formula for the general term.

Find the General Term (nth Term) of an Arithmetic Sequence

In the following exercises, find the term described using the information provided.

Find the eleventh term of a sequence where the third term is 19 and the common difference is five. Give the formula for the general term.

Find the General Term (nth Term) of an Arithmetic Sequence

In the following exercises, find the term described using the information provided.

Find the eighth term of a sequence where the seventh term is $\mathrm{-8}$ and the common difference is $\mathrm{-5}.$ Give the formula for the general term.

In the following exercises, find the first term and common difference of the sequence with the given terms. Give the formula for the general term.

The second term is 14 and the thirteenth term is 47.

In the following exercises, find the first term and common difference of the sequence with the given terms. Give the formula for the general term.

The second term is 13 and the tenth term is $\mathrm{-51}.$

In the following exercises, find the first term and common difference of the sequence with the given terms. Give the formula for the general term.

The fourth term is $\mathrm{-6}$ and the fifteenth term is 27.

Find the Sum of the First n Terms of an Arithmetic Sequence

In the following exercises, find the sum of the first 30 terms of each arithmetic sequence.

$11,$ $14,$ $17,$ $20,$ $23\text{,}\text{…}$

Find the Sum of the First n Terms of an Arithmetic Sequence

In the following exercises, find the sum of the first 30 terms of each arithmetic sequence.

$8,$ $5,$ $2,$ $\mathrm{-1},$ $\mathrm{-4}\text{,}\text{…}$

Find the Sum of the First n Terms of an Arithmetic Sequence

In the following exercises, find the sum of the first 30 terms of each arithmetic sequence.

$\mathrm{-17},$ $\mathrm{-15},$ $\mathrm{-13},$ $\mathrm{-11},$ $\mathrm{-9}\text{,}\text{…}$

In the following exercises, find the sum of the first 50 terms of the arithmetic sequence whose general term is given.

$a_n=5n-1$

In the following exercises, find the sum of the first 50 terms of the arithmetic sequence whose general term is given.

$a_n=\mathrm{-3}n+5$

In the following exercises, find each sum.

${\displaystyle \sum _{i=1}^{40}(8i-7})$

In the following exercises, find each sum.

${\displaystyle \sum _{i=1}^{50}(3i+6})$

In the following exercises, find each sum.

${\displaystyle \sum _{i=1}^{35}(\mathrm{-6}i}-2)$

In the following exercises, find each sum.

In your own words, explain how to determine whether a sequence is arithmetic.

In the following exercises, find each sum.

In your own words, explain how to find the general term of an arithmetic sequence.

Simplify: $\frac{24}{32}.$
If you missed this problem, review Example 1.24.

Evaluate: ⓐ $3^4$ ⓑ ${\left(\frac{1}{2}\right)}^4.$
If you missed this problem, review Example 1.19.

If $f\left(x\right)=4·3^x,$ find ⓐ $f\left(1\right)$ ⓑ $f\left(2\right)$ ⓒ $f\left(3\right).$
If you missed this problem, review Example 3.49.

Determine if each sequence is geometric. If so indicate the common ratio.

ⓐ $7,$ $21,$ $63,$ $189,$ $567,$ $\mathrm{1,701}\text{,}\text{…}$

ⓑ $64,$ $16,$ $4,$ $1,$ $\frac{1}{4},$ $\frac{1}{16}\text{,}\text{…}$

ⓒ $2,$ $4,$ $12,$ $48,$ $240,$ $\mathrm{1,440}\text{,}\text{…}$

Determine if each sequence is geometric. If so indicate the common ratio.

ⓐ $\mathrm{-150},$ $\mathrm{-30},$ $\mathrm{-15},$ $\mathrm{-5},$ $-$ $\frac{5}{2},$ $0\text{,}\text{…}$

ⓑ $5,$ $10,$ $20,$ $40,$ $80,$ $160\text{,}\text{…}$

ⓒ $8,$ $4,$ $2,$ $1,$ $\frac{1}{2},$ $\frac{1}{4}\text{,}\text{…}$

Write the first five terms of the sequence where the first term is 7 and the common ratio is $r=\mathrm{-3}.$

Write the first five terms of the sequence where the first term is 6 and the common ratio is $r=\mathrm{-4}.$

Find the thirteenth term of a sequence where the first term is 81 and the common ratio is $r=\frac{1}{3}.$

Find the twelfth term of a sequence where the first term is 256 and the common ratio is $r=\frac{1}{4}.$

Find the ninth term of the sequence 6, 18, 54, 162, 486, 1,458, … Then find the general term for the sequence.

Find the eleventh term of the sequence 7, 14, 28, 56, 112, 224, … Then find the general term for the sequence.

Find the sum of the first 20 terms of the geometric sequence 3, 6, 12, 24, 48, 96, …

Find the sum of the first 20 terms of the geometric sequence 6, 18, 54, 162, 486, 1,458, …

Find the sum: ${\displaystyle \sum _{i=1}^{15}6{\left(2\right)}^i}.$

Find the sum: ${\displaystyle \sum _{i=1}^{10}5{\left(2\right)}^i}.$

Find the sum of the infinite geometric series $48+$ $24+$ $12+$ $6+$ $3+$ $\frac{3}{2}+$ $\dots$

Find the sum of the infinite geometric series $64+$ $16+$ $4+$ $1+$ $\frac{1}{4}+$ $\frac{1}{16}+$ $\dots$

Write the repeating decimal $0.\bar{4}$ as a fraction.

Write the repeating decimal $0.\bar{8}$ as a fraction.

What is the total effect on the economy of a government tax rebate of $1,000 to each household in order to stimulate the economy if each household will spend 90% of the rebate in goods and services?

What is the total effect on the economy of a government tax rebate of $500 to each household in order to stimulate the economy if each household will spend 85% of the rebate in goods and services?

New grandparents decide to invest $200 per month in an annuity for their grandson. The account will pay 5% interest per year which is compounded monthly. How much will be in the child’s account at his twenty-first birthday?

Arturo just got his first full-time job after graduating from college at age 27. He decided to invest $200 per month in an IRA (an annuity). The interest on the annuity is 8%, which is compounded monthly. How much will be in the Arturo’s account when he retires at his sixty-seventh birthday?

Determine if a Sequence is Geometric

In the following exercises, determine if the sequence is geometric, and if so, indicate the common ratio.

$3,$ $12,$ $48,$ $192,$ $768,$ $3072\text{,}\text{…}$

Determine if a Sequence is Geometric

In the following exercises, determine if the sequence is geometric, and if so, indicate the common ratio.

$72,$ $36,$ $18,$ $9,$ $\frac{9}{2}\text{,}\frac{9}{4}\text{,}\text{…}$

Determine if a Sequence is Geometric

In the following exercises, determine if the sequence is geometric, and if so, indicate the common ratio.

$\mathrm{-3},$ $6,$ $\mathrm{-12},$ $24,$ $\mathrm{-48},$ $96\text{,}\text{…}$

In the following exercises, determine if each sequence is arithmetic, geometric or neither. If arithmetic, indicate the common difference. If geometric, indicate the common ratio.

$48,$ $24,$ $12,$ $6,$ $3,$ $\frac{3}{2}\text{,}\text{…}$

In the following exercises, determine if each sequence is arithmetic, geometric or neither. If arithmetic, indicate the common difference. If geometric, indicate the common ratio.

$\mathrm{-7},$ $\mathrm{-2},$ $3,$ $8,$ $13,$ $18\text{,}\text{…}$

In the following exercises, determine if each sequence is arithmetic, geometric or neither. If arithmetic, indicate the common difference. If geometric, indicate the common ratio.

$\frac{1}{2},$ $\frac{1}{4},$ $\frac{1}{8},$ $\frac{1}{16},$ $\frac{1}{32},$ $\frac{1}{64}\text{,}\text{…}$

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

$a_1=4$ and $r=3$

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

$a_1=\mathrm{-4}$ and $r=\mathrm{-2}$

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

$a_1=27$ and $r=\frac{1}{3}$

Find the General Term (nth Term) of a Geometric Sequence

In the following exercises, find the indicated term of a sequence where the first term and the common ratio is given.

Find $a_{11}$ given $a_1=8$ and $r=3.$

Find the General Term (nth Term) of a Geometric Sequence

In the following exercises, find the indicated term of a sequence where the first term and the common ratio is given.

Find $a_{10}$ given $a_1=\mathrm{-6}$ and $r=\mathrm{-2}.$

Find the General Term (nth Term) of a Geometric Sequence

In the following exercises, find the indicated term of a sequence where the first term and the common ratio is given.

Find $a_{10}$ given $a_1=\mathrm{100,000}$ and $r=0.1.$

In the following exercises, find the indicated term of the given sequence. Find the general term for the sequence.

Find $a_9$ of the sequence, $9,$ $18,$ $36,$ $72,$ $144,$ $288\text{,}\text{…}$

In the following exercises, find the indicated term of the given sequence. Find the general term for the sequence.

Find $a_{15}$ of the sequence, $\mathrm{-486},$ $162,$ $\mathrm{-54},$ $18,$ $\mathrm{-6},$ $2\text{,}\text{…}$

In the following exercises, find the indicated term of the given sequence. Find the general term for the sequence.

Find $a_{10}$ of the sequence, $1,$ $0.1,$ $0.01,$ $0.001,$ $0.0001,$ $0.00001\text{,}\text{…}$

Find the Sum of the First n terms of a Geometric Sequence

In the following exercises, find the sum of the first fifteen terms of each geometric sequence.

$8,$ $24,$ $72,$ $216,$ $648,$ $1944\text{,}\text{…}$

Find the Sum of the First n terms of a Geometric Sequence

In the following exercises, find the sum of the first fifteen terms of each geometric sequence.

$\mathrm{-6},$ $12,$ $\mathrm{-24},$ $48,$ $\mathrm{-96},$ $192\text{,}\text{…}$

Find the Sum of the First n terms of a Geometric Sequence

In the following exercises, find the sum of the first fifteen terms of each geometric sequence.

$81,$ $27,$ $9,$ $3,$ $1,$ $\frac{1}{3}\text{,}\text{…}$

In the following exercises, find the sum of the geometric sequence.

${\displaystyle \sum _{i=1}^{15}{\left(2\right)}^i}$

In the following exercises, find the sum of the geometric sequence.

${\displaystyle \sum _{i=1}^{9}4{\left(2\right)}^i}$

In the following exercises, find the sum of the geometric sequence.

${\displaystyle \sum _{i=1}^{10}9{\left(\frac{1}{3}\right)}^i}$

Find the Sum of an Infinite Geometric Series

In the following exercises, find the sum of each infinite geometric series.

$1+$ $\frac{1}{3}+$ $\frac{1}{9}+$ $\frac{1}{27}+$ $\frac{1}{81}+$ $\frac{1}{243}+$ $\frac{1}{729}+$ $\dots$

Find the Sum of an Infinite Geometric Series

In the following exercises, find the sum of each infinite geometric series.

$6-$ $2+$ $\frac{2}{3}-$ $\frac{2}{9}+$ $\frac{2}{27}-$ $\frac{2}{81}+$ $\dots$

Find the Sum of an Infinite Geometric Series

In the following exercises, find the sum of each infinite geometric series.

$6+$ $12+$ $24+$ $48+$ $96+$ $192+$ $\dots$

Find the Sum of an Infinite Geometric Series

In the following exercises, find the sum of each infinite geometric series.

$\mathrm{1,024}+$ $512+$ $256+$ $128+$ $64+$ $32+$ $\dots$

In the following exercises, write each repeating decimal as a fraction.

$0.\bar{3}$

In the following exercises, write each repeating decimal as a fraction.

$0.\bar{7}$

In the following exercises, write each repeating decimal as a fraction.

$0.\stackrel{—}{45}$

Apply Geometric Sequences and Series in the Real World

In the following exercises, solve the problem.

Find the total effect on the economy of each government tax rebate to each household in order to stimulate the economy if each household will spend the indicated percent of the rebate in goods and services.

	Tax rebate to each household	Percent spent on goods and services	Total Effect on the economy
ⓐ	$\text{\$}\mathrm{1,000}$	$85\text{\%}$
ⓑ	$\text{\$}\mathrm{1,000}$	$75\text{\%}$
ⓒ	$\text{\$}\mathrm{1,500}$	$90\text{\%}$
ⓓ	$\text{\$}\mathrm{1,500}$	$80\text{\%}$