Series and Their Notations: Apply It 2

Annuities

An annuity is an investment in which the purchaser makes a sequence of periodic, equal payments. To find the amount of an annuity, we need to find the sum of all the payments and the interest earned.

A parent invests [latex]$50[/latex] each month into a college fund. The account paid [latex]6\%[/latex] annual interest, compounded monthly. How much money will be accumulated in [latex]6[/latex] years?

To find the interest rate per payment period, we need to divide the [latex]6\%[/latex] annual percentage interest (APR) rate by [latex]12[/latex]. So, the monthly interest rate is [latex]0.5\%[/latex]. We can multiply the amount in the account each month by [latex]100.5 \%[/latex] to find the value of the account after interest has been added. We can find the value of the annuity right after the last deposit by using a geometric series with [latex]{a}_{1}=50[/latex] and [latex]r=100.5 \%=1.005[/latex].

  • After the first deposit, the value of the annuity will be [latex]$50[/latex].

Let us see if we can determine the amount in the college fund and the interest earned. We can find the value of the annuity after [latex]n[/latex] deposits using the formula for the sum of the first [latex]n[/latex] terms of a geometric series. In [latex]6[/latex] years, there are [latex]72[/latex] months, so [latex]n=72[/latex]. We can substitute [latex]{a}_{1}=50, r=1.005,[/latex] and [latex]n=72[/latex] into the formula, and simplify to find the value of the annuity after [latex]6[/latex] years.

[latex]{S}_{72}=\dfrac{50\left(1-{1.005}^{72}\right)}{1 - 1.005}\approx 4\text{,}320.44[/latex]

After the last deposit, the couple will have a total of [latex]$4,320.44[/latex] in the account.

Notice, the couple made [latex]72[/latex] payments of [latex]$50[/latex] each for a total of [latex]72\left(50\right) = $3,600[/latex].

This means that because of the annuity, the couple earned [latex]$720.44[/latex] interest in their college fund.

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

  1. Determine [latex]{a}_{1}[/latex], the value of the initial deposit.
  2. Determine [latex]n[/latex], the number of deposits.
  3. Determine [latex]r[/latex].
    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 [latex]r[/latex].
  4. Substitute values for [latex]{a}_{1},r,[/latex] and [latex]n[/latex]
    into the formula for the sum of the first [latex]n[/latex] terms of a geometric series, [latex]{S}_{n}=\dfrac{{a}_{1}\left(1-{r}^{n}\right)}{1-r}[/latex].
  5. Simplify to find [latex]{S}_{n}[/latex], the value of the annuity after [latex]n[/latex] deposits.
A deposit of [latex]$100[/latex] is placed into a college fund at the beginning of every month for [latex]10[/latex] years. The fund earns [latex]9 \%[/latex] annual interest, compounded monthly, and paid at the end of the month. How much is in the account right after the last deposit?

At the beginning of each month, [latex]$200[/latex] is deposited into a retirement fund. The fund earns [latex]6 \%[/latex] 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 [latex]10[/latex] years?