# 13.4 Series and their notations  (Page 3/18)

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A man earns $100 in the first week of June. Each week, he earns$12.50 more than the previous week. After 12 weeks, how much has he earned?

$2,025 ## Using the formula for geometric series Just as the sum of the terms of an arithmetic sequence is called an arithmetic series, the sum of the terms in a geometric sequence is called a geometric series . Recall that a geometric sequence is a sequence in which the ratio of any two consecutive terms is the common ratio , $\text{\hspace{0.17em}}r.\text{\hspace{0.17em}}$ We can write the sum of the first $n$ terms of a geometric series as ${S}_{n}={a}_{1}+r{a}_{1}+{r}^{2}{a}_{1}+...+{r}^{n–1}{a}_{1}.$ Just as with arithmetic series, we can do some algebraic manipulation to derive a formula for the sum of the first $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ terms of a geometric series. We will begin by multiplying both sides of the equation by $\text{\hspace{0.17em}}r.\text{\hspace{0.17em}}$ $r{S}_{n}=r{a}_{1}+{r}^{2}{a}_{1}+{r}^{3}{a}_{1}+...+{r}^{n}{a}_{1}$ Next, we subtract this equation from the original equation. Notice that when we subtract, all but the first term of the top equation and the last term of the bottom equation cancel out. To obtain a formula for ${S}_{n},$ divide both sides by $\left(1-r\right).$ ## Formula for the sum of the first n Terms of a geometric series A geometric series is the sum of the terms in a geometric sequence. The formula for the sum of the first $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ terms of a geometric sequence is represented as Given a geometric series, find the sum of the first n terms. 1. Identify $\text{\hspace{0.17em}}{a}_{1},\text{\hspace{0.17em}}r,\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}n.$ 2. Substitute values for $\text{\hspace{0.17em}}{a}_{1},\text{\hspace{0.17em}}r,$ and $n$ into the formula ${S}_{n}=\frac{{a}_{1}\left(1–{r}^{n}\right)}{1–r}.$ 3. Simplify to find ${S}_{n}.$ ## Finding the first n Terms of a geometric series Use the formula to find the indicated partial sum of each geometric series. 1. ${S}_{11}$ for the series 2. $\underset{6}{\overset{k=1}{{\sum }^{\text{​}}}}3\cdot {2}^{k}$ 1. ${a}_{1}=8,$ and we are given that $n=11.$ We can find $r$ by dividing the second term of the series by the first. $r=\frac{-4}{8}=-\frac{1}{2}$ Substitute values for into the formula and simplify. $\begin{array}{l}{S}_{n}=\frac{{a}_{1}\left(1-{r}^{n}\right)}{1-r}\hfill \\ {S}_{11}=\frac{8\left(1-{\left(-\frac{1}{2}\right)}^{11}\right)}{1-\left(-\frac{1}{2}\right)}\approx 5.336\hfill \end{array}$ 2. Find ${a}_{1}$ by substituting $k=1$ into the given explicit formula. ${a}_{1}=3\cdot {2}^{1}=6$ We can see from the given explicit formula that $r=2.$ The upper limit of summation is 6, so $n=6.$ Substitute values for ${a}_{1},\text{\hspace{0.17em}}r,$ and $n$ into the formula, and simplify. $\begin{array}{l}{S}_{n}=\frac{{a}_{1}\left(1-{r}^{n}\right)}{1-r}\hfill \\ {S}_{6}=\frac{6\left(1-{2}^{6}\right)}{1-2}=378\hfill \end{array}$ Use the formula to find the indicated partial sum of each geometric series. ${S}_{20}$ for the series $\approx 2,000.00$ $\sum _{k=1}^{8}{3}^{k}$ 9,840 ## Solving an application problem with a geometric series At a new job, an employee’s starting salary is$26,750. He receives a 1.6% annual raise. Find his total earnings at the end of 5 years.

The problem can be represented by a geometric series with ${a}_{1}=26,750\text{;}\text{\hspace{0.17em}}$ $n=5\text{;}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}r=1.016.$ Substitute values for $\text{\hspace{0.17em}}{a}_{1}\text{,}\text{\hspace{0.17em}}$ $r\text{,}$ and $n$ into the formula and simplify to find the total amount earned at the end of 5 years.

$\begin{array}{l}{S}_{n}=\frac{{a}_{1}\left(1-{r}^{n}\right)}{1-r}\hfill \\ {S}_{5}=\frac{26\text{,}750\left(1-{1.016}^{5}\right)}{1-1.016}\approx 138\text{,}099.03\hfill \end{array}$

He will have earned a total of $138,099.03 by the end of 5 years. At a new job, an employee’s starting salary is$32,100. She receives a 2% annual raise. How much will she have earned by the end of 8 years?

\$275,513.31

## Using the formula for the sum of an infinite geometric series

Thus far, we have looked only at finite series. Sometimes, however, we are interested in the sum of the terms of an infinite sequence rather than the sum of only the first $n$ terms. An infinite series    is the sum of the terms of an infinite sequence. An example of an infinite series is $2+4+6+8+...$

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