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.

ⓐ Write a formula for the child's weekly allowance in a given year.

ⓑ What will the child's allowance be when he is 16 years old?

Solution

ⓐ  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

ⓑ  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(11)=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?
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