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13.4 Series and their notations  (Page 6/18)

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We can find the value of the annuity after n deposits using the formula for the sum of the first n terms of a geometric series. In 6 years, there are 72 months, so n = 72. We can substitute a 1 = 50 ,   r = 1.005 ,   and   n = 72 into the formula, and simplify to find the value of the annuity after 6 years.

S 72 = 50 ( 1 − 1.005 72 ) 1 − 1.005 ≈ 4 , 320.44

After the last deposit, the couple will have a total of $4,320.44 in the account. Notice, the couple made 72 payments of $50 each for a total of   72(50) = $3,600 .   This means that because of the annuity, the couple earned $720.44 interest in their college fund.

Given an initial deposit and an interest rate, find the value of an annuity.

  1. Determine   a 1 ,   the value of the initial deposit.
  2. Determine   n ,   the number of deposits.
  3. Determine   r .
    1. Divide the annual interest rate by the number of times per year that interest is compounded.
    2. Add 1 to this amount to find r .
  4. Substitute values for   a 1 ,   r ,   and   n   into the formula for the sum of the first n terms of a geometric series, S n = a 1 ( 1 – r n ) 1 – r .
  5. Simplify to find S n , the value of the annuity after n deposits.

Solving an annuity problem

A deposit of $100 is placed into a college fund at the beginning of every month for 10 years. The fund earns 9% annual interest, compounded monthly, and paid at the end of the month. How much is in the account right after the last deposit?

The value of the initial deposit is $100, so   a 1 = 100.   A total of 120 monthly deposits are made in the 10 years, so n = 120. To find r ,   divide the annual interest rate by 12 to find the monthly interest rate and add 1 to represent the new monthly deposit.

r = 1 + 0.09 12 = 1.0075

Substitute   a 1 = 100 ,   r = 1.0075 ,   and   n = 120   into the formula for the sum of the first n terms of a geometric series, and simplify to find the value of the annuity.

S 120 = 100 ( 1 − 1.0075 120 ) 1 − 1.0075 ≈ 19 , 351.43

So the account has $19,351.43 after the last deposit is made.

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At the beginning of each month, $200 is deposited into a retirement fund. The fund earns 6% annual interest, compounded monthly, and paid into the account at the end of the month. How much is in the account if deposits are made for 10 years?

$92,408.18

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Access these online resources for additional instruction and practice with series.

Key equations

sum of the first   n   terms of an arithmetic series S n = n ( a 1 + a n ) 2
sum of the first   n   terms of a geometric series S n = a 1 ( 1 − r n ) 1 − r ⋅ r ≠ 1
sum of an infinite geometric series with   – 1 < r <   1 S n = a 1 1 − r ⋅ r ≠ 1

Key concepts

  • The sum of the terms in a sequence is called a series.
  • A common notation for series is called summation notation, which uses the Greek letter sigma to represent the sum. See [link] .
  • The sum of the terms in an arithmetic sequence is called an arithmetic series.
  • The sum of the first n terms of an arithmetic series can be found using a formula. See [link] and [link] .
  • The sum of the terms in a geometric sequence is called a geometric series.
  • The sum of the first n terms of a geometric series can be found using a formula. See [link] and [link] .
  • The sum of an infinite series exists if the series is geometric with –1 < r < 1.
  • If the sum of an infinite series exists, it can be found using a formula. See [link] , [link] , and [link] .
  • An annuity is an account into which the investor makes a series of regularly scheduled payments. The value of an annuity can be found using geometric series. See [link] .

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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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