# 9.2 Arithmetic sequences  (Page 4/8)

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Find the number of terms in the finite arithmetic sequence.

There are 11 terms in the sequence.

## 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.

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?
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. 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. ## Key equations  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 $\begin{array}{l}{a}_{n}={a}_{1}+d\left(n-1\right)\end{array}$ ## Key concepts • 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 ${a}_{n}={a}_{1}+d\left(n-1\right).$ 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. 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. #### Questions & Answers hello, I am happy to help! Shirley Reply please can go further on polynomials quadratic Abdullahi I need quadratic equation link to Alpa Beta Abdullahi Reply find the value of 2x=32 Felix Reply divide by 2 on each side of the equal sign to solve for x corri X=16 Michael Want to review on complex number 1.What are complex number 2.How to solve complex number problems. Beyan use the y -intercept and slope to sketch the graph of the equation y=6x Only Reply how do we prove the quadratic formular Seidu Reply hello, if you have a question about Algebra 2. I may be able to help. I am an Algebra 2 Teacher Shirley Reply thank you help me with how to prove the quadratic equation Seidu may God blessed u for that. Please I want u to help me in sets. Opoku what is math number Tric Reply 4 Trista x-2y+3z=-3 2x-y+z=7 -x+3y-z=6 Sidiki Reply Need help solving this problem (2/7)^-2 Simone Reply x+2y-z=7 Sidiki what is the coefficient of -4× Mehri Reply -1 Shedrak the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1 Alfred Reply An investment account was opened with an initial deposit of$9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
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12, 17, 22.... 25th term
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