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# Finding the Number of Terms in a Finite Arithmetic Sequence

Explicit formulas can be used to determine the number of terms in a finite arithmetic sequence. We need to find the common difference, and then determine how many times the common difference must be added to the first term to obtain the final term of the sequence.

### How To: Given the first three terms and the last term of a finite arithmetic sequence, find the total number of terms.

1. Find the common difference $d$.
2. Substitute the common difference and the first term into ${a}_{n}={a}_{1}+d\left(n - 1\right)$.
3. Substitute the last term for ${a}_{n}$ and solve for $n$.

### Example 6: Finding the Number of Terms in a Finite Arithmetic Sequence

Find the number of terms in the finite arithmetic sequence.
$\left\{8\text{, }1\text{, }-6\text{, }...\text{, }-41\right\}$

### Solution

The common difference can be found by subtracting the first term from the second term.
$1 - 8=-7$
The common difference is $-7$ . Substitute the common difference and the initial term of the sequence into the $n\text{th}$ term formula and simplify.
$\begin{array}{l}{a}_{n}={a}_{1}+d\left(n - 1\right)\hfill \\ {a}_{n}=8+-7\left(n - 1\right)\hfill \\ {a}_{n}=15 - 7n\hfill \end{array}$
Substitute $-41$ for ${a}_{n}$ and solve for $n$
$\begin{array}{l}-41=15 - 7n\hfill \\ 8=n\hfill \end{array}$
There are eight terms in the sequence.

### Try It 7

Find the number of terms in the finite arithmetic sequence.
$\left\{6\text{, }11\text{, }16\text{, }...\text{, }56\right\}$
Solution

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

### Example 7: 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?

### Solution

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.

### Try It 8

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? Solution