<< Chapter < Page Chapter >> Page >

Evaluate k = 2 5 ( 3 k 1 ) .

38

Got questions? Get instant answers now!

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
Got questions? Get instant answers now!
Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

13 + 21 + 29 +  + 69

328

Got questions? Get instant answers now!

k = 1 10 5 6 k

−280

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask