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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] .

Questions & Answers

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The denominator of a certain fraction is 9 more than the numerator. If 6 is added to both terms of the fraction, the value of the fraction becomes 2/3. Find the original fraction. 2. The sum of the least and greatest of 3 consecutive integers is 60. What are the valu
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1. x + 6 2 -------------- = _ x + 9 + 6 3 x + 6 3 ----------- x -- (cross multiply) x + 15 2 3(x + 6) = 2(x + 15) 3x + 18 = 2x + 30 (-2x from both) x + 18 = 30 (-18 from both) x = 12 Test: 12 + 6 18 2 -------------- = --- = --- 12 + 9 + 6 27 3
Pawel
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Mark = x,. Don = 3x + 1 x + 3x + 1 = 113 4x = 112, x = 28 Mark = 28, Don = 85, 28 + 85 = 113
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Solve for the first variable in one of the equations, then substitute the result into the other equation. Point For: (6111,4111,−411)(6111,4111,-411) Equation Form: x=6111,y=4111,z=−411x=6111,y=4111,z=-411
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Source:  OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3
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