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.
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?
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.
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.
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
${a}_{n}={a}_{1}+d(n-1).$ 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] .
Section exercises
Verbal
What is an arithmetic sequence?
A sequence where each successive term of the sequence increases (or decreases) by a constant value.
At the start of a trip, the odometer on a car read 21,395. At the end of the trip, 13.5 hours later, the odometer read 22,125. Assume the scale on the odometer is in miles. What is the average speed the car traveled during this trip?