### Review Exercises

##### Sequences

**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.

${a}_{n}=7n-5$

${a}_{n}={2}^{n}+n$

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

**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.

$\mathrm{-5},\mathrm{-4},\mathrm{-3},\mathrm{-2},\mathrm{-1},\text{\u2026}$

$1,\mathrm{-8},27,\mathrm{-64},125,\text{\u2026}$

**Use Factorial Notation**

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

${a}_{n}=4n!$

${a}_{n}=\frac{(n-1)!}{{(n+1)}^{\text{2}}}$

**Find the Partial Sum**

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

$\sum _{i=1}^{3}{5}^{i}$

$\sum _{k=1}^{4}(k+1)\left(2k+1\right)$

**Use Summation Notation to write a Sum**

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

$4-8+12-16+20-24$

##### Arithmetic Sequences

**Determine if a Sequence is Arithmetic**

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

$1,2,4,8,16,32,\text{\u2026}$

$13,9,5,1,\mathrm{-3},\mathrm{-7},\text{\u2026}$

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

${a}_{1}=8$ and $d=\mathrm{-2}$

**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-fifth term of a sequence where the first term is five and the common difference is three.

Find the thirtieth term of a sequence where the first term is 16 and the common difference is $\mathrm{-5}.$

Find the seventeenth term of a sequence where the first term is $\mathrm{-21}$ and the common difference is two.

In the following exercises, find the indicated term and give the formula for the general term.

Find the eighteenth term of a sequence where the fifth term is $12$ and the common difference is seven.

Find the twenty-first term of a sequence where the seventh term is $14$ and the common difference is $\mathrm{-3}.$

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 third term is $\mathrm{-26}$ and the sixteenth term is $\mathrm{-91}.$

**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.

$1,6,11,16,21,\text{\u2026}$

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

${a}_{n}=\mathrm{-2}n+19$

In the following exercises, find each sum.

$\sum _{i=1}^{30}(\mathrm{-3}i-7})$

##### Geometric Sequences and Series

**Determine if a Sequence is Geometric**

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

$3,12,48,192,768,3072,\text{\u2026}$

$112,56,28,14,7,\frac{7}{2},\text{\u2026}$

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

${a}_{1}=\mathrm{-3}$ and $r=5$

${a}_{1}=5$ and $r=\mathrm{-3}$

**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 ${a}_{1}=\mathrm{10,000,000}$ and $r=0.1.$

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

Find ${a}_{12}$ of the sequence, $6,\mathrm{-24},96,\mathrm{-384},1536,\mathrm{-6144},\text{\u2026}$

Find ${a}_{9}$ of the sequence, $4374,1458,486,162,54,18,\text{\u2026}$

**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.

$3,12,48,192,768,3072\text{\u2026}$

In the following exercises, find the sum

$\sum _{i=1}^{8}7{\left(3\right)}^{i}$

**Find the Sum of an Infinite Geometric Series**

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

$1-\frac{1}{3}+\frac{1}{9}-\frac{1}{27}+\frac{1}{81}-\frac{1}{243}+\frac{1}{729}-\text{\u2026}$

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

$0.\stackrel{\u2013}{8}$

**Apply Geometric Sequences and Series in the Real World**

In the following exercises, solve the problem.

What is the total effect on the economy of a government tax rebate of $\text{\$}360$ to each household in order to stimulate the economy if each household will spend $60\%$ of the rebate in goods and services?

Adam just got his first full-time job after graduating from high school at age 17. He decided to invest $\text{\$}300$ per month in an IRA (an annuity). The interest on the annuity is $7\%$ which is compounded monthly. How much will be in Adam’s account when he retires at his sixty-seventh birthday?

##### Binomial Theorem

**Use Pascal’s Triangle to Expand a Binomial**

In the following exercises, expand each binomial using Pascal’s Triangle.

${\left(a+b\right)}^{7}$

${\left(x+6\right)}^{3}$

${\left(7x+2y\right)}^{3}$

**Evaluate a Binomial Coefficient**

In the following exercises, evaluate.

ⓐ $\left(\begin{array}{c}11\hfill \\ 1\hfill \end{array}\right)$

ⓑ $\left(\begin{array}{c}12\hfill \\ 12\hfill \end{array}\right)$

ⓒ $\left(\begin{array}{c}13\hfill \\ 0\hfill \end{array}\right)$

ⓓ $\left(\begin{array}{c}8\hfill \\ 3\hfill \end{array}\right)$

ⓐ $\left(\begin{array}{c}7\hfill \\ 1\hfill \end{array}\right)$

ⓑ $\left(\begin{array}{c}5\hfill \\ 5\hfill \end{array}\right)$

ⓒ $\left(\begin{array}{c}9\hfill \\ 0\hfill \end{array}\right)$

ⓓ $\left(\begin{array}{c}9\hfill \\ 5\hfill \end{array}\right)$

ⓐ $\left(\begin{array}{c}1\hfill \\ 1\hfill \end{array}\right)$

ⓑ $\left(\begin{array}{c}15\hfill \\ 15\hfill \end{array}\right)$

ⓒ $\left(\begin{array}{c}4\hfill \\ 0\hfill \end{array}\right)$

ⓓ $\left(\begin{array}{c}11\hfill \\ 2\hfill \end{array}\right)$

**Use the Binomial Theorem to Expand a Binomial**

In the following exercises, expand each binomial, using the Binomial Theorem.

${\left(p+q\right)}^{6}$

${\left(2x+1\right)}^{4}$

${\left(x-3y\right)}^{5}$

In the following exercises, find the indicated term in the expansion of the binomial.

Third term of ${\left(x-y\right)}^{7}$

In the following exercises, find the coefficient of the indicated term in the expansion of the binomial.

${x}^{5}$ term of ${\left(x-2\right)}^{8}$