Find the regular interval between terms in a simple sequence and use it to write the sequence’s terms
Use recursive and explicit formulas to represent and analyze arithmetic sequences
Solving Application Problems with Arithmetic Sequences
In many application problems, it often makes sense to use an initial term of [latex]{a}_{0}[/latex] instead of [latex]{a}_{1}[/latex]. In these problems we alter the explicit formula slightly to account for the difference in initial terms. We use the following formula:
[latex]{a}_{n}={a}_{0}+dn[/latex]
A five-year old child receives an allowance of [latex]$1[/latex] each week. His parents promise him an annual increase of [latex]$2[/latex] 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 [latex]16[/latex] years old?
The situation can be modeled by an arithmetic sequence with an initial term of [latex]1[/latex] and a common difference of [latex]2[/latex]. Let [latex]A[/latex] be the amount of the allowance and [latex]n[/latex] be the number of years after age [latex]5[/latex]. Using the altered explicit formula for an arithmetic sequence we get:
[latex]{A}_{n}=1+2n[/latex]
We can find the number of years since age [latex]5[/latex] by subtracting.
[latex]16 - 5=11[/latex]
We are looking for the child’s allowance after [latex]11[/latex] years. Substitute [latex]11[/latex] into the formula to find the child’s allowance at age [latex]16[/latex].
[latex]{A}_{11}=1+2\left(11\right)=23[/latex]
The child’s allowance at age [latex]16[/latex] will be [latex]$23[/latex] per week.
A woman decides to go for a [latex]10[/latex]-minute run every day this week and plans to increase the time of her daily run by [latex]4[/latex] minutes each week. Write a formula for the time of her run after [latex]n[/latex] weeks. How long will her daily run be [latex]8[/latex] weeks from today?
The formula is [latex]{T}_{n}=10+4n[/latex], and it will take her [latex]42[/latex] minutes.