1. Learning objectives

Learning objectives

Review the key ideas of this chapter.

Apply the chapter's methods to review exercises.

Check your understanding with a practice test.

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2. Worked example

Worked example

annuity

An annuity is an investment that is a sequence of equal periodic deposits.

arithmetic sequence

An arithmetic sequence is a sequence where the difference between consecutive terms is constant.

common difference

d

n

common ratio

r

n

finite sequence

A sequence with a domain that is limited to a finite number of counting numbers.

general term of a sequence

n

geometric sequence

A geometric sequence is a sequence where the ratio between consecutive terms is always the same

infinite geometric series

An infinite geometric series is an infinite sum infinite geometric sequence.

infinite sequence

A sequence whose domain is all counting numbers and there is an infinite number of counting numbers.

partial sum

When we add a finite number of terms of a sequence, we call the sum a partial sum.

sequence

A sequence is a function whose domain is the counting numbers.

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Key Concepts

Factorial Notation If n is a positive integer, then $n!$ is

n!=n(n-1)(n-2)\ldots(3)(2)(1).

We define $0!$ as 1, so $0!=1$

Summation Notation The sum of the first n terms of a sequence whose *n*th term ${a}_{n}$ is written in summation notation as:

\overset{n}{\underset{i=1}{\sum}}{a}_{i}={a}_{1}+{a}_{2}+{a}_{3}+{a}_{4}+{a}_{5}+\ldots+{a}_{n}

The i is the index of summation and the 1 tells us where to start and the n tells us where to end.

General Term (*n*th term) of an Arithmetic Sequence The general term of an arithmetic sequence with first term ${a}_{1}$ and the common difference d is

{a}_{n}={a}_{1}+(n-1)d

Sum of the First n Terms of an Arithmetic Sequence The sum, ${S}_{n},$ of the first n terms of an arithmetic sequence, where ${a}_{1}$ is the first term and ${a}_{n}$ is the *n*th term is

{S}_{n}=\frac{n}{2}({a}_{1}+{a}_{n})

General Term (*n*th term) of a Geometric Sequence: The general term of a geometric sequence with first term ${a}_{1}$ and the common ratio r is

{a}_{n}={a}_{1}{r}^{n-1}

Sum of the First n Terms of a Geometric Series: The sum, ${S}_{n},$ of the n terms of a geometric sequence is

{S}_{n}=\frac{{a}_{1}(1-{r}^{n})}{1-r}

where ${a}_{1}$ is the first term and r is the common ratio.

Infinite Geometric Series: An infinite geometric series is an infinite sum whose first term is ${a}_{1}$ and common ratio is r and is written

{a}_{1}+{a}_{1}r+{a}_{1}{r}^{2}+\ldots+{a}_{1}{r}^{n-1}+\ldots

Sum of an Infinite Geometric Series: For an infinite geometric series whose first term is ${a}_{1}$ and common ratio r, $\begin{array}{l} \,\mathrm{If}\,|r|<1,\,\mathrm{the sum is} \\ \,{S}_{}=\frac{{a}_{1}}{1-r} \\ \,\mathrm{We say the series converges.} \\ \,\mathrm{If}\,|r|\ge{}1,\,\mathrm{the infinite geometric series does not have a sum. We say the series diverges.} \\ \end{array}$

Value of an Annuity with Interest Compounded $n$ Times a Year: For a principal, P, invested at the end of a compounding period, with an interest rate, r, which is compounded n times a year, the new balance, A, after t years, is

{A}_{t}=\frac{P({(1+\frac{r}{n})}^{nt}-1)}{\frac{r}{n}}

Patterns in the expansion of ${(a+b)}^{n}$ The number of terms is $n+1.$ The first term is ${a}^{n}$ and the last term is ${b}^{n}.$ The exponents on a decrease by one on each term going left to right. The exponents on b increase by one on each term going left to right. The sum of the exponents on any term is n.

Pascal’s Triangle

Binomial Coefficient $(\begin{array}{l} n \\ r \\ \end{array})$ : A binomial coefficient $(\begin{array}{l} n \\ r \\ \end{array}),$ where r and n are integers with $0\le{}r\le{}n,$ is defined as

(\begin{array}{l} n \\ r \\ \end{array})=\frac{n!}{r!(n-r)!}

We read $(\begin{array}{l} n \\ r \\ \end{array})$ as “n choose r” or “n taken r at a time”.

Properties of Binomial Coefficients $\begin{array}{ccccccccc} (\begin{array}{l} n \\ 1 \\ \end{array})=n & & & & (\begin{array}{l} n \\ n \\ \end{array})=1 & & & & (\begin{array}{l} n \\ 0 \\ \end{array})=1 \\ \end{array}$

Binomial Theorem: For any real numbers a, b, and positive integer n,

{(a+b)}^{n}=(\begin{array}{l} n \\ 0 \\ \end{array}){a}^{n}+(\begin{array}{l} n \\ 1 \\ \end{array}){a}^{n-1}{b}^{1}+(\begin{array}{l} n \\ 2 \\ \end{array}){a}^{n-2}{b}^{2}+...+(\begin{array}{l} n \\ r \\ \end{array}){a}^{n-r}{b}^{r}+...+(\begin{array}{l} n \\ n \\ \end{array}){b}^{n}

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Review Exercises

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.

Find a Formula for the General Term (*n*th Term) of a Sequence

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

Use Factorial Notation

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

Find the Partial Sum

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

Use Summation Notation to write a Sum

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

Determine if a Sequence is Arithmetic

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

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

Find the General Term (*n*th Term) of an Arithmetic Sequence

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

In the following exercises, find the indicated term and 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.

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.

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

In the following exercises, find each sum.

Determine if a Sequence is Geometric

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

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

Find the General Term (*n*th 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.

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

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.

In the following exercises, find the sum

Find the Sum of an Infinite Geometric Series

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

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

Apply Geometric Sequences and Series in the Real World

In the following exercises, solve the problem.

Use Pascal’s Triangle to Expand a Binomial

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

Evaluate a Binomial Coefficient

In the following exercises, evaluate.

Use the Binomial Theorem to Expand a Binomial

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

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

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

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