Use the formula for the sum of the first n terms of an arithmetic series.
Use the formula for the sum of the first n terms of a geometric series.
Use the formula for the sum of an infinite geometric series.
Solve word problems involving series.
Solving Annuity Problems
Consider a couple who invested a set amount of money each month into a college fund for six years.
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.
Suppose the couple invests $50 each month. This is the value of the initial deposit. The account paid 6% annual interest, compounded monthly. To find the interest rate per payment period, we need to divide the 6% annual percentage interest (APR) rate by 12. So the monthly interest rate is 0.5%. We can multiply the amount in the account each month by 100.5% 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 $50. 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 6 years, there are 72 months, so [latex]n=72[/latex]. We can substitute [latex]{a}_{1}=50, r=1.005, \text{and} n=72[/latex] 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 [latex]72\left(50\right) = $3,600[/latex]. This means that because of the annuity, the couple earned $720.44 interest in their college fund.
How To: Given an initial deposit and an interest rate, find the value of an annuity.
Determine [latex]{a}_{1}[/latex], the value of the initial deposit.
Determine [latex]n[/latex], the number of deposits.
Determine [latex]r[/latex].
Divide the annual interest rate by the number of times per year that interest is compounded.
Add 1 to this amount to find [latex]r[/latex].
Substitute values for [latex]{a}_{1}\text{,}r,\text{ and }n[/latex]
into the formula for the sum of the first [latex]n[/latex] terms of a geometric series, [latex]{S}_{n}=\frac{{a}_{1}\left(1-{r}^{n}\right)}{1-r}[/latex].
Simplify to find [latex]{S}_{n}[/latex], the value of the annuity after [latex]n[/latex] deposits.
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 [latex]{a}_{1}=100[/latex]. A total of 120 monthly deposits are made in the 10 years, so [latex]n=120[/latex]. To find [latex]r[/latex], divide the annual interest rate by 12 to find the monthly interest rate and add 1 to represent the new monthly deposit.
[latex]r=1+\frac{0.09}{12}=1.0075[/latex]
Substitute [latex]{a}_{1}=100\text{,}r=1.0075\text{,}\text{and}n=120[/latex] into the formula for the sum of the first [latex]n[/latex] terms of a geometric series, and simplify to find the value of the annuity.