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Find the number of terms in the finite arithmetic sequence.

{ 6 11 16 ... 56 }

There are 11 terms in the sequence.

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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 + d n

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 + 2 n
  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 ( 11 ) = 23

    The child’s allowance at age 16 will be $23 per week.

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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 + 4 n , and it will take her 42 minutes.

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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 n 2
explicit formula for nth term of an arithmetic sequence a n = a 1 + d ( n 1 )

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 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 + d n . 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.

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How is the common difference of an arithmetic sequence found?

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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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Practice Key Terms 2

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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