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Evaluate k = 2 5 ( 3 k 1 ) .

38

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Using the formula for arithmetic series

Just as we studied special types of sequences, we will look at special types of series. Recall that an arithmetic sequence    is a sequence in which the difference between any two consecutive terms is the common difference    , d . The sum of the terms of an arithmetic sequence is called an arithmetic series . We can write the sum of the first n terms of an arithmetic series as:

S n = a 1 + ( a 1 + d ) + ( a 1 + 2 d ) + ... + ( a n d ) + a n .

We can also reverse the order of the terms and write the sum as

S n = a n + ( a n d ) + ( a n 2 d ) + ... + ( a 1 + d ) + a 1 .

If we add these two expressions for the sum of the first n terms of an arithmetic series, we can derive a formula for the sum of the first n terms of any arithmetic series.

S n = a 1 + ( a 1 + d ) + ( a 1 + 2 d ) + ... + ( a n d ) + a n + S n = a n + ( a n d ) + ( a n 2 d ) + ... + ( a 1 + d ) + a 1 2 S n = ( a 1 + a n ) + ( a 1 + a n ) + ... + ( a 1 + a n )

Because there are n terms in the series, we can simplify this sum to

2 S n = n ( a 1 + a n ) .

We divide by 2 to find the formula for the sum of the first n terms of an arithmetic series.

S n = n ( a 1 + a n ) 2

Formula for the sum of the first n Terms of an arithmetic series

An arithmetic series    is the sum of the terms of an arithmetic sequence. The formula for the sum of the first n terms of an arithmetic sequence is

S n = n ( a 1 + a n ) 2

Given terms of an arithmetic series, find the sum of the first n terms.

  1. Identify a 1 and a n .
  2. Determine n .
  3. Substitute values for a 1 a n , and n into the formula S n = n ( a 1 + a n ) 2 .
  4. Simplify to find S n .

Finding the first n Terms of an arithmetic series

Find the sum of each arithmetic series.

  1. 5 + 8 + 11 + 14 + 17 + 20 + 23 + 26 + 29 + 32
  2. 20 + 15 + 10 +…+ −50
  3. k = 1 12 3 k 8
  1. We are given a 1 = 5 and a n = 32.

    Count the number of terms in the sequence to find n = 10.

    Substitute values for a 1 , a n  , and n into the formula and simplify.

      S n = n ( a 1 + a n ) 2 S 10 = 10 ( 5 + 32 ) 2 = 185
  2. We are given a 1 = 20 and a n = 50.

    Use the formula for the general term of an arithmetic sequence to find n .

    a n = a 1 + ( n 1 ) d 50 = 20 + ( n 1 ) ( 5 ) 70 = ( n 1 ) ( 5 ) 14 = n 1 15 = n

    Substitute values for a 1 , a n , n into the formula and simplify.

    S n = n ( a 1 + a n ) 2 S 15 = 15 ( 20 50 ) 2 = 225
  3. To find a 1 , substitute k = 1 into the given explicit formula.

    a k = 3 k 8   a 1 = 3 ( 1 ) 8 = 5

    We are given that n = 12. To find a 12 , substitute k = 12 into the given explicit formula.

      a k = 3 k 8 a 12 = 3 ( 12 ) 8 = 28

    Substitute values for a 1 , a n , and n into the formula and simplify.

      S n = n ( a 1 + a n ) 2 S 12 = 12 ( 5 + 28 ) 2 = 138
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Use the formula to find the sum of each arithmetic series.

1 .4 + 1 .6 + 1 .8 + 2 .0 + 2 .2 + 2 .4 + 2 .6 + 2 .8 + 3 .0 + 3 .2 + 3 .4

26 .4

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13 + 21 + 29 +  + 69

328

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k = 1 10 5 6 k

−280

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Solving application problems with arithmetic series

On the Sunday after a minor surgery, a woman is able to walk a half-mile. Each Sunday, she walks an additional quarter-mile. After 8 weeks, what will be the total number of miles she has walked?

This problem can be modeled by an arithmetic series with a 1 = 1 2 and d = 1 4 . We are looking for the total number of miles walked after 8 weeks, so we know that n = 8 , and we are looking for S 8 . To find a 8 , we can use the explicit formula for an arithmetic sequence.

a n = a 1 + d ( n 1 ) a 8 = 1 2 + 1 4 ( 8 1 ) = 9 4

We can now use the formula for arithmetic series.

  S n = n ( a 1 + a n ) 2    S 8 = 8 ( 1 2 + 9 4 ) 2 = 11

She will have walked a total of 11 miles.

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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