# 11.4 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 $2+4+6+8+...$

what is set?
a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
I got 300 minutes. is it right?
Patience
no. should be about 150 minutes.
Jason
It should be 158.5 minutes.
Mr
ok, thanks
Patience
100•3=300 300=50•2^x 6=2^x x=log_2(6) =2.5849625 so, 300=50•2^2.5849625 and, so, the # of bacteria will double every (100•2.5849625) = 258.49625 minutes
Thomas
what is the importance knowing the graph of circular functions?
can get some help basic precalculus
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
can get some help inverse function
ismail
Rectangle coordinate
how to find for x
it depends on the equation
Robert
yeah, it does. why do we attempt to gain all of them one side or the other?
Melissa
whats a domain
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
Spiro; thanks for putting it out there like that, 😁
Melissa
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
difference between calculus and pre calculus?
give me an example of a problem so that I can practice answering
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
I want to learn about the law of exponent
explain this
what is functions?
A mathematical relation such that every input has only one out.
Spiro
yes..it is a relationo of orders pairs of sets one or more input that leads to a exactly one output.
Mubita
Is a rule that assigns to each element X in a set A exactly one element, called F(x), in a set B.
RichieRich