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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 ${a}_{1}=50,r=1.005,\text{and}n=72$ 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 $\text{\hspace{0.17em}}\text{72(50)=\$3,600}\text{.}\text{\hspace{0.17em}}$ 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.
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 $\text{\hspace{0.17em}}{a}_{1}=100.\text{\hspace{0.17em}}$ A total of 120 monthly deposits are made in the 10 years, so $n=120.$ To find $r,\text{\hspace{0.17em}}$ divide the annual interest rate by 12 to find the monthly interest rate and add 1 to represent the new monthly deposit.
Substitute $\text{\hspace{0.17em}}{a}_{1}=100\text{,}\text{\hspace{0.17em}}r=1.0075\text{,}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}n=120\text{\hspace{0.17em}}$ into the formula for the sum of the first $n$ terms of a geometric series, and simplify to find the value of the annuity.
So the account has $19,351.43 after the last deposit is made.
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
Access these online resources for additional instruction and practice with series.
sum of the first $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ terms of an arithmetic series | ${S}_{n}=\frac{n({a}_{1}+{a}_{n})}{2}$ |
sum of the first $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ terms of a geometric series | $${S}_{n}=\frac{{a}_{1}(1-{r}^{n})}{1-r}\cdot r\ne 1$$ |
sum of an infinite geometric series with $\text{\hspace{0.17em}}\u20131<r<\text{}1$ | $${S}_{n}=\frac{{a}_{1}}{1-r}\cdot r\ne 1$$ |
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