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Find the number of terms in the finite arithmetic sequence.
There are 11 terms in the sequence.
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 five-year old child receives an allowance of $1 each week. His parents promise him an annual increase of $2 per week.
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:
We can find the number of years since age 5 by subtracting.
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.
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.
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 | $\begin{array}{l}{a}_{n}={a}_{1}+d(n-1)\end{array}$ |
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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