13.4 Series and their notations  (Page 6/18)

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 $a_1=50, r=1.005, \text{and} n=72$ into the formula, and simplify to find the value of the annuity after 6 years.

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 $\text{72(50) = \$3,600}\text{.}$ This means that because of the annuity, the couple earned $720.44 interest in their college fund.

Key equations

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 $\mathrm{–1}<r<1.$
• 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] .