# 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] .

A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
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