# 9.4 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 $2+4+6+8+...$ #### Questions & Answers what is math number Tric Reply x-2y+3z=-3 2x-y+z=7 -x+3y-z=6 Sidiki Reply Need help solving this problem (2/7)^-2 Simone Reply x+2y-z=7 Sidiki what is the coefficient of -4× Mehri Reply -1 Shedrak the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1 Alfred Reply An investment account was opened with an initial deposit of$9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
Kala Reply
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
Moses Reply
12, 17, 22.... 25th term
Alexandra Reply
12, 17, 22.... 25th term
Akash
College algebra is really hard?
Shirleen Reply
Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table.
Carole
I'm 13 and I understand it great
AJ
I am 1 year old but I can do it! 1+1=2 proof very hard for me though.
Atone
hi
Adu
Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily.
Vedant
hi vedant can u help me with some assignments
Solomon
find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
Abhi
If f(x) = x-2 then, f(3) when 5f(x+1) 5((3-2)+1) 5(1+1) 5(2) 10
Augustine
how do they get the third part x = (32)5/4
kinnecy Reply
make 5/4 into a mixed number, make that a decimal, and then multiply 32 by the decimal 5/4 turns out to be
AJ
how
Sheref
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
salma
But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends
salma
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
hi
salma
hi
Ayuba
Hello
opoku
hi
Ali
greetings from Iran
Ali
salut. from Algeria
Bach
hi
Nharnhar

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Source:  OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3
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