# 13.4 Series and their notations  (Page 6/18)

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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 into the formula, and simplify to find the value of the annuity after 6 years.

${S}_{72}=\frac{50\left(1-{1.005}^{72}\right)}{1-1.005}\approx 4\text{,}320.44$

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 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 $\text{\hspace{0.17em}}{a}_{1}\text{,}\text{\hspace{0.17em}}$ the value of the initial deposit. 2. Determine $\text{\hspace{0.17em}}n\text{,}\text{\hspace{0.17em}}$ the number of deposits. 3. Determine $\text{\hspace{0.17em}}r.$ 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 $\text{\hspace{0.17em}}{a}_{1}\text{,}\text{\hspace{0.17em}}r,\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ into the formula for the sum of the first $n$ terms of a geometric series, ${S}_{n}=\frac{{a}_{1}\left(1–{r}^{n}\right)}{1–r}.$ 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 $\text{\hspace{0.17em}}{a}_{1}=100.\text{\hspace{0.17em}}$ 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. $r=1+\frac{0.09}{12}=1.0075$ 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. ${S}_{120}=\frac{100\left(1-{1.0075}^{120}\right)}{1-1.0075}\approx 19\text{,}351.43$ 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.

## Key equations

 sum of the first $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ terms of an arithmetic series ${S}_{n}=\frac{n\left({a}_{1}+{a}_{n}\right)}{2}$ sum of the first $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ terms of a geometric series ${S}_{n}=\frac{{a}_{1}\left(1-{r}^{n}\right)}{1-r}\cdot r\ne 1$ sum of an infinite geometric series with ${S}_{n}=\frac{{a}_{1}}{1-r}\cdot r\ne 1$

## 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
• 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] .

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