12.2 Arithmetic Sequences

Learning objectives

Learning objectives

• Determine if a sequence is arithmetic

• Find the general term ( $n$ th term) of an arithmetic sequence

• Find the sum of the first $n$ terms of an arithmetic sequence

Determine if a Sequence is Arithmetic

The last section introduced sequences and now we will look at two specific types of sequences that each have special properties. In this section we will look at arithmetic sequences and in the next section, geometric sequences.

An arithmetic sequence is a sequence where the difference between consecutive terms is constant. The difference between consecutive terms in an arithmetic sequence, ${a}_{n}-{a}_{n-1},$ is d, the common difference, for n greater than or equal to two.

In each of these sequences, the difference between consecutive terms is constant, and so the sequence is arithmetic.

Example 12.13

Solution

To determine if the sequence is arithmetic, we find the difference of the consecutive terms shown.

ⓐ

Find the difference ofthe consecutive terms. · $\begin{array}{ccccccccccc} 5, & & \,9, & & \,13, & & \,17 & & \,21, & \,25,\ldots \\ 9-5 & & \,13-9 & & \,17-13 & & \,21-17 & & \,25-21 & & \\ 4 & & \,4 & & \,4 & & \,4 & & \,4 & & \\ \end{array}$

The sequence is arithmetic. The common difference is $d=4.$

ⓑ

Find the difference ofthe consecutive terms. · $\begin{array}{ccccccccccc} 4, & & \,9, & & \,12, & & \,17 & & \,20, & \,25,\ldots \\ 9-4 & & \,12-9 & & \,17-12 & & \,20-17 & & \,25-20 & & \\ 5 & & \,3 & & \,5 & & \,3 & & \,5 & & \\ \end{array}$

The sequence is not arithmetic as all the differences betweenthe consecutive terms are not the same.There is no common difference.

ⓒ

Find the difference ofthe consecutive terms. · $\begin{array}{ccccccccccc} 10, & & \,3, & & \,-4, & & \,-11 & & \,-18, & \,-25,\ldots & \\ 3-10 & & \,-4-3 & & \,-11-(-4) & & \,-18-(-11) & & \,-25-(-18) & & \\ -7 & & \,-7 & & \,-7 & & \,-7 & & \,-7 & & \\ \end{array}$

The sequence is arithmetic. The common difference is $d=-7.$

If we know the first term, ${a}_{1},$ and the common difference, d, we can list a finite number of terms of the sequence.

Example 12.14

Solution

We start with the first term and add the common difference. Then we add the common difference to that result to get the next term, and so on.

The sequence is $5,-1,-7,-13,-19,\,\ldots$

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

Just as we found a formula for the general term of a sequence, we can also find a formula for the general term of an arithmetic sequence.

Let’s write the first few terms of a sequence where the first term is ${a}_{1}$ and the common difference is d. We will then look for a pattern.

As we look for a pattern we see that each term starts with ${a}_{1}$.

The first term adds 0*d* to the ${a}_{1}$, the second term adds 1*d*, the third term adds 2*d*, the fourth term adds 3*d*, and the fifth term adds 4*d*. The number of *d*s that were added to ${a}_{1}$ is one less than the number of the term. This leads us to the following

We will use this formula in the next example to find the $15^{th}$ term of a sequence.

Example 12.15

Solution

To find the fifteenth term, ${a}_{15},$ use theformula with ${a}_{1}=3\,\mathrm{and}\,d=6.$ · $\,{a}_{n}={a}_{1}+(n-1)d$ Substitute in the values. · $\,{a}_{15}=3+(15-1)6$ Simplify. · $\,{a}_{15}=3+(14)6$

$\,{a}_{15}=87$

Sometimes we do not know the first term and we must use other given information to find it before we find the requested term.

Example 12.16

Solution

To first find the first term, ${a}_{1},$ use theformula with ${a}_{7}=10,n=7,\,\mathrm{and}\,d=-2.$ · $\,{a}_{n}={a}_{1}+(n-1)d$ Substitute in the values. · $\,10={a}_{1}+(7-1)(-2)$ Simplify. · $\,10={a}_{1}+(6)(-2)$

$\,10={a}_{1}-12$

$\,{a}_{1}=22$

Find the twelfth term, ${a}_{12},$ using theformula with ${a}_{1}=22,n=12,\,\mathrm{and}\,d=-2.$ · $\,{a}_{n}={a}_{1}+(n-1)d$ Substitute in the values. · $\,{a}_{12}=22+(12-1)(-2)$ Simplify. · $\,{a}_{12}=22+(11)(-2)$

$\,{a}_{12}=0$

$\,$The twelfth term of the sequence is 0, ${a}_{12}=0.$

To find the general term, substitutethe values into the formula. · $\,{a}_{n}={a}_{1}+(n-1)d$

$\,{a}_{n}=22+(n-1)(-2)$

$\,{a}_{n}=22-2n+2$

$\,$The general term is ${a}_{n}=-2n+24.$

Sometimes the information given leads us to two equations in two unknowns. We then use our methods for solving systems of equations to find the values needed.

Example 12.17

Solution

Since we know two terms, we can make a system of equations using the formula for the general term.

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

As with the general sequences, it is often useful to find the sum of an arithmetic sequence. The sum, ${S}_{n},$ of the first $n$ terms of any arithmetic sequence is written as ${S}_{n}={a}_{1}+{a}_{2}+{a}_{3}+...+{a}_{n}.$ To find the sum by merely adding all the terms can be tedious. So we can also develop a formula to find the sum of a sequence using the first and last term of the sequence.

We can develop this new formula by first writing the sum by starting with the first term, ${a}_{1},$ and keep adding a d to get the next term as:

We can also reverse the order of the terms and write the sum by starting with ${a}_{n}$ and keep subtracting d to get the next term as

If we add these two expressions for the sum of the first n terms of an arithmetic sequence, we can derive a formula for the sum of the first n terms of any arithmetic series.

Because there are n sums of $({a}_{1}+{a}_{n})$ on the right side of the equation, we rewrite the right side as $n({a}_{1}+{a}_{n}).$

We divide by two to solve for ${S}_{n}.$

This gives us a general formula for the sum of the first n terms of an arithmetic sequence.

We apply this formula in the next example where the first few terms of the sequence are given.

Example 12.18

Solution

To find the sum, we will use the formula ${S}_{n}=\frac{n}{2}({a}_{1}+{a}_{n}).$ We know ${a}_{1}=8$, $d=5$ and $n=30,$ but we need to find ${a}_{n}$ in order to use the sum formula.

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

$\mathrm{Find}\,{a}_{n}\,\mathrm{where}\,{a}_{1}=8,d=5\,\mathrm{and}\,n=30.$ · ${a}_{30}=8+(30-1)5$ Simplify. · ${a}_{30}=8+(29)5$

${a}_{30}=153$

$\mathrm{Knowing}\,{a}_{1}=8,n=30,\,\mathrm{and}\,{a}_{30}=153,$use the sum formula. · $\,{S}_{n}=\frac{n}{2}({a}_{1}+{a}_{n})$ Substitute in the values. · ${S}_{30}=\frac{30}{2}(8+153)$ Simplify. · ${S}_{30}=15(161)$ Simplify. · ${S}_{30}=2,415$

In the next example, we are given the general term for the sequence and are asked to find the sum of the first 50 terms.

Example 12.19

Solution

To find the sum, we will use the formula ${S}_{n}=\frac{n}{2}({a}_{1}+{a}_{n}).$ We know $n=50,$ but we need to find ${a}_{1}$ and ${a}_{n}$ in order to use the sum formula.

In the next example we are given the sum in summation notation. To add all the terms would be tedious, so we extract the information needed to use the formula to find the sum of the first n terms.

Example 12.20

Solution

To find the sum, we will use the formula ${S}_{n}=\frac{n}{2}({a}_{1}+{a}_{n}).$ We know $n=25,$ but we need to find ${a}_{1}$ and ${a}_{n}$ in order to use the sum formula.

Section 12.2 Exercises

Section 12.2 Exercises

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 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 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 50 terms of the arithmetic sequence whose general term is given.

In the following exercises, find each sum.

Writing Exercises

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ After reviewing this checklist, what will you do to become confident for all objectives?

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

ⓐ $9,20,31,42,53,64,\,\ldots$ ⓑ $12,6,0,-6,-12,-18,\,\ldots$ ⓒ $7,1,10,4,13,7,\,\ldots$

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

ⓐ $-4,4,2,10,8,16,\,\ldots$ ⓑ $-3,-1,1,3,5,7,\,\ldots$ ⓒ $7,2,-3,-8,-13,-18,\,\ldots$

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

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

2. Find the eighteenth term of a sequence where the first term is 13 and the common difference is $-7$.

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

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

2. 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.

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

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

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

3. Find the sum of the first 50 terms of the arithmetic sequence whose general term is ${a}_{n}=2n-5.$

1. Find the sum of the first 50 terms of the arithmetic sequence whose general term is ${a}_{n}=4n+3.$

2. Find the sum: $\overset{30}{\underset{i=1}{\sum}}(6i-4).$

3. Find the sum: $\overset{35}{\underset{i=1}{\sum}}(5i-3).$

1. $4,12,20,28,36,44,\,\ldots$

2. $-15,-16,3,12,21,30,\,\ldots$

3. $8,5,2,-1,-4,-7,\,\ldots$

1. ${a}_{1}=11$ and $d=7$

2. ${a}_{1}=-7$ and $d=4$

3. ${a}_{1}=14$ and $d=-9$

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

2. Find the thirtieth term of a sequence where the first term is $-14$ and the common difference is five.

3. Find the sixteenth term of a sequence where the first term is 11 and the common difference is $-6.$

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

2. 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.

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

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

2. The second term is 13 and the tenth term is $-51.$

3. The fourth term is $-6$ and the fifteenth term is 27.

1. $11,14,17,20,23,\,\ldots$

2. $8,5,2,-1,-4,\,\ldots$

3. ${a}_{n}=5n-1$

1. ${a}_{n}=-3n+5$

2. $\overset{40}{\underset{i=1}{\sum}}(8i-7)$

3. $\overset{50}{\underset{i=1}{\sum}}(3i+6)$

1. $\overset{35}{\underset{i=1}{\sum}}(-6i-2)$

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

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