# 13.2 Arithmetic sequences  (Page 4/8)

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Find the number of terms in the finite arithmetic sequence.

There are 11 terms in the sequence.

## Solving application problems with arithmetic sequences

In many application problems, it often makes sense to use an initial term of ${a}_{0}$ instead of ${a}_{1}.$ In these problems, we alter the explicit formula slightly to account for the difference in initial terms. We use the following formula:

${a}_{n}={a}_{0}+dn$

## Solving application problems with arithmetic sequences

A five-year old child receives an allowance of $1 each week. His parents promise him an annual increase of$2 per week.

1. Write a formula for the child’s weekly allowance in a given year.
2. What will the child’s allowance be when he is 16 years old?
1. The situation can be modeled by an arithmetic sequence with an initial term of 1 and a common difference of 2.

Let $A$ be the amount of the allowance and $n$ be the number of years after age 5. Using the altered explicit formula for an arithmetic sequence we get:

${A}_{n}=1+2n$
2. We can find the number of years since age 5 by subtracting.

$16-5=11$

We are looking for the child’s allowance after 11 years. Substitute 11 into the formula to find the child’s allowance at age 16.

${A}_{11}=1+2\left(11\right)=23$

The child’s allowance at age 16 will be \$23 per week.

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?

The formula is ${T}_{n}=10+4n,\text{\hspace{0.17em}}$ and it will take her 42 minutes.

Access this online resource for additional instruction and practice with arithmetic sequences.

## Key equations

 recursive formula for nth term of an arithmetic sequence ${a}_{n}={a}_{n-1}+d\phantom{\rule{1}{0ex}}n\ge 2$ explicit formula for nth term of an arithmetic sequence $\begin{array}{l}{a}_{n}={a}_{1}+d\left(n-1\right)\end{array}$

## Key concepts

• An arithmetic sequence is a sequence where the difference between any two consecutive terms is a constant.
• The constant between two consecutive terms is called the common difference.
• The common difference is the number added to any one term of an arithmetic sequence that generates the subsequent term. See [link] .
• The terms of an arithmetic sequence can be found by beginning with the initial term and adding the common difference repeatedly. See [link] and [link] .
• A recursive formula for an arithmetic sequence with common difference $d$ is given by ${a}_{n}={a}_{n-1}+d,n\ge 2.$ See [link] .
• As with any recursive formula, the initial term of the sequence must be given.
• An explicit formula for an arithmetic sequence with common difference $d$ is given by ${a}_{n}={a}_{1}+d\left(n-1\right).$ See [link] .
• An explicit formula can be used to find the number of terms in a sequence. See [link] .
• In application problems, we sometimes alter the explicit formula slightly to ${a}_{n}={a}_{0}+dn.$ See [link] .

## Verbal

What is an arithmetic sequence?

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

How is the common difference of an arithmetic sequence found?

How do we determine whether a sequence is arithmetic?

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.

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