# 11.4 Series and their notations  (Page 7/18)

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## Verbal

What is an $n\text{th}$ partial sum?

An $n\text{th}$ partial sum is the sum of the first $n$ terms of a sequence.

What is the difference between an arithmetic sequence and an arithmetic series?

What is a geometric series?

A geometric series is the sum of the terms in a geometric sequence.

How is finding the sum of an infinite geometric series different from finding the $n\text{th}$ partial sum?

What is an annuity?

An annuity is a series of regular equal payments that earn a constant compounded interest.

## Algebraic

For the following exercises, express each description of a sum using summation notation.

The sum of terms ${m}^{2}+3m$ from $m=1$ to $m=5$

The sum from of $n=0$ to $n=4$ of $5n$

$\sum _{n=0}^{4}5n$

The sum of $6k-5$ from $k=-2$ to $k=1$

The sum that results from adding the number 4 five times

$\sum _{k=1}^{5}4$

For the following exercises, express each arithmetic sum using summation notation.

$5+10+15+20+25+30+35+40+45+50$

$10+18+26+\dots +162$

$\sum _{k=1}^{20}8k+2$

$\frac{1}{2}+1+\frac{3}{2}+2+\dots +4$

For the following exercises, use the formula for the sum of the first $n$ terms of each arithmetic sequence.

$\frac{3}{2}+2+\frac{5}{2}+3+\frac{7}{2}$

${S}_{5}=\frac{5\left(\frac{3}{2}+\frac{7}{2}\right)}{2}$

$19+25+31+\dots +73$

$3.2+3.4+3.6+\dots +5.6$

${S}_{13}=\frac{13\left(3.2+5.6\right)}{2}$

For the following exercises, express each geometric sum using summation notation.

$1+3+9+27+81+243+729+2187$

$8+4+2+\dots +0.125$

$\sum _{k=1}^{7}8\cdot {0.5}^{k-1}$

$-\frac{1}{6}+\frac{1}{12}-\frac{1}{24}+\dots +\frac{1}{768}$

For the following exercises, use the formula for the sum of the first $n$ terms of each geometric sequence, and then state the indicated sum.

$9+3+1+\frac{1}{3}+\frac{1}{9}$

${S}_{5}=\frac{9\left(1-{\left(\frac{1}{3}\right)}^{5}\right)}{1-\frac{1}{3}}=\frac{121}{9}\approx 13.44$

$\sum _{n=1}^{9}5\cdot {2}^{n-1}$

$\sum _{a=1}^{11}64\cdot {0.2}^{a-1}$

${S}_{11}=\frac{64\left(1-{0.2}^{11}\right)}{1-0.2}=\frac{781,249,984}{9,765,625}\approx 80$

For the following exercises, determine whether the infinite series has a sum. If so, write the formula for the sum. If not, state the reason.

$12+18+24+30+...$

$2+1.6+1.28+1.024+...$

The series is defined. $S=\frac{2}{1-0.8}$

$\sum _{m=1}^{\infty }{4}^{m-1}$

$\underset{\infty }{\overset{k=1}{{\sum }^{\text{​}}}}-{\left(-\frac{1}{2}\right)}^{k-1}$

The series is defined. $S=\frac{-1}{1-\left(-\frac{1}{2}\right)}$

## Graphical

For the following exercises, use the following scenario. Javier makes monthly deposits into a savings account. He opened the account with an initial deposit of $50. Each month thereafter he increased the previous deposit amount by$20.

Graph the arithmetic sequence showing one year of Javier’s deposits.

Graph the arithmetic series showing the monthly sums of one year of Javier’s deposits.

For the following exercises, use the geometric series ${\sum _{k=1}^{\infty }\left(\frac{1}{2}\right)}^{k}.$

Graph the first 7 partial sums of the series.

What number does ${S}_{n}$ seem to be approaching in the graph? Find the sum to explain why this makes sense.

Sample answer: The graph of ${S}_{n}$ seems to be approaching 1. This makes sense because $\sum _{k=1}^{\infty }{\left(\frac{1}{2}\right)}^{k}$ is a defined infinite geometric series with $S=\frac{\frac{1}{2}}{1–\left(\frac{1}{2}\right)}=1.$

## Numeric

For the following exercises, find the indicated sum.

$\sum _{a=1}^{14}a$

$\sum _{n=1}^{6}n\left(n-2\right)$

49

$\sum _{k=1}^{17}{k}^{2}$

$\sum _{k=1}^{7}{2}^{k}$

254

For the following exercises, use the formula for the sum of the first $n$ terms of an arithmetic series to find the sum.

$-1.7+-0.4+0.9+2.2+3.5+4.8$

$6+\frac{15}{2}+9+\frac{21}{2}+12+\frac{27}{2}+15$

${S}_{7}=\frac{147}{2}$

$-1+3+7+...+31$

$\sum _{k=1}^{11}\left(\frac{k}{2}-\frac{1}{2}\right)$

${S}_{11}=\frac{55}{2}$

For the following exercises, use the formula for the sum of the first $n$ terms of a geometric series to find the partial sum.

${S}_{6}$ for the series $-2-10-50-250...$

${S}_{7}$ for the series $0.4-2+10-50...$

${S}_{7}=5208.4$

$\sum _{k=1}^{9}{2}^{k-1}$

$\sum _{n=1}^{10}-2\cdot {\left(\frac{1}{2}\right)}^{n-1}$

${S}_{10}=-\frac{1023}{256}$

For the following exercises, find the sum of the infinite geometric series.

$4+2+1+\frac{1}{2}...$

$-1-\frac{1}{4}-\frac{1}{16}-\frac{1}{64}...$

$S=-\frac{4}{3}$

$\underset{\infty }{\overset{k=1}{{\sum }^{\text{​}}}}3\cdot {\left(\frac{1}{4}\right)}^{k-1}$

$\sum _{n=1}^{\infty }4.6\cdot {0.5}^{n-1}$

$S=9.2$

For the following exercises, determine the value of the annuity for the indicated monthly deposit amount, the number of deposits, and the interest rate.

Deposit amount: $\text{}50;$ total deposits: $60;$ interest rate: $5%,$ compounded monthly

Deposit amount: $\text{}150;$ total deposits: $24;$ interest rate: $3%,$ compounded monthly

$3,705.42 Deposit amount: $\text{}450;$ total deposits: $60;$ interest rate: $4.5%,$ compounded quarterly Deposit amount: $\text{}100;$ total deposits: $120;$ interest rate: $10%,$ compounded semi-annually$695,823.97

## Extensions

The sum of terms $50-{k}^{2}$ from $k=x$ through $7$ is $115.$ What is x ?

Write an explicit formula for ${a}_{k}$ such that $\sum _{k=0}^{6}{a}_{k}=189.$ Assume this is an arithmetic series.

${a}_{k}=30-k$

Find the smallest value of n such that $\sum _{k=1}^{n}\left(3k–5\right)>100.$

How many terms must be added before the series has a sum less than $-75?$

9 terms

Write $0.\overline{65}$ as an infinite geometric series using summation notation. Then use the formula for finding the sum of an infinite geometric series to convert $0.\overline{65}$ to a fraction.

The sum of an infinite geometric series is five times the value of the first term. What is the common ratio of the series?

$r=\frac{4}{5}$

To get the best loan rates available, the Riches want to save enough money to place 20% down on a $160,000 home. They plan to make monthly deposits of$125 in an investment account that offers 8.5% annual interest compounded semi-annually. Will the Riches have enough for a 20% down payment after five years of saving? How much money will they have saved?

Karl has two years to save $10,000$ to buy a used car when he graduates. To the nearest dollar, what would his monthly deposits need to be if he invests in an account offering a 4.2% annual interest rate that compounds monthly?

$400 per month ## Real-world applications Keisha devised a week-long study plan to prepare for finals. On the first day, she plans to study for $\text{\hspace{0.17em}}1\text{\hspace{0.17em}}$ hour, and each successive day she will increase her study time by $\text{\hspace{0.17em}}30\text{\hspace{0.17em}}$ minutes. How many hours will Keisha have studied after one week? A boulder rolled down a mountain, traveling 6 feet in the first second. Each successive second, its distance increased by 8 feet. How far did the boulder travel after 10 seconds? 420 feet A scientist places 50 cells in a petri dish. Every hour, the population increases by 1.5%. What will the cell count be after 1 day? A pendulum travels a distance of 3 feet on its first swing. On each successive swing, it travels $\frac{3}{4}$ the distance of the previous swing. What is the total distance traveled by the pendulum when it stops swinging? 12 feet Rachael deposits$1,500 into a retirement fund each year. The fund earns 8.2% annual interest, compounded monthly. If she opened her account when she was 19 years old, how much will she have by the time she is 55? How much of that amount will be interest earned?

#### Questions & Answers

I've run into this: x = r*cos(angle1 + angle2) Which expands to: x = r(cos(angle1)*cos(angle2) - sin(angle1)*sin(angle2)) The r value confuses me here, because distributing it makes: (r*cos(angle2))(cos(angle1) - (r*sin(angle2))(sin(angle1)) How does this make sense? Why does the r distribute once
How can you tell what type of parent function a graph is ?
generally by how the graph looks and understanding what the base parent functions look like and perform on a graph
William
if you have a graphed line, you can have an idea by how the directions of the line turns, i.e. negative, positive, zero
William
y=x will obviously be a straight line with a zero slope
William
y=x^2 will have a parabolic line opening to positive infinity on both sides of the y axis vice versa with y=-x^2 you'll have both ends of the parabolic line pointing downward heading to negative infinity on both sides of the y axis
William
y=x will be a straight line, but it will have a slope of one. Remember, if y=1 then x=1, so for every unit you rise you move over positively one unit. To get a straight line with a slope of 0, set y=1 or any integer.
Aaron
yes, correction on my end, I meant slope of 1 instead of slope of 0
William
what is f(x)=
I don't understand
Joe
Typically a function 'f' will take 'x' as input, and produce 'y' as output. As 'f(x)=y'. According to Google, "The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain."
Thomas
Sorry, I don't know where the "Â"s came from. They shouldn't be there. Just ignore them. :-)
Thomas
Darius
Thanks.
Thomas
Â
Thomas
It is the Â that should not be there. It doesn't seem to show if encloses in quotation marks. "Â" or 'Â' ... Â
Thomas
Now it shows, go figure?
Thomas
what is this?
i do not understand anything
unknown
lol...it gets better
Darius
I've been struggling so much through all of this. my final is in four weeks 😭
Tiffany
this book is an excellent resource! have you guys ever looked at the online tutoring? there's one that is called "That Tutor Guy" and he goes over a lot of the concepts
Darius
thank you I have heard of him. I should check him out.
Tiffany
is there any question in particular?
Joe
I have always struggled with math. I get lost really easy, if you have any advice for that, it would help tremendously.
Tiffany
Sure, are you in high school or college?
Darius
Hi, apologies for the delayed response. I'm in college.
Tiffany
how to solve polynomial using a calculator
So a horizontal compression by factor of 1/2 is the same as a horizontal stretch by a factor of 2, right?
The center is at (3,4) a focus is at (3,-1), and the lenght of the major axis is 26
The center is at (3,4) a focus is at (3,-1) and the lenght of the major axis is 26 what will be the answer?
Rima
I done know
Joe
What kind of answer is that😑?
Rima
I had just woken up when i got this message
Joe
Can you please help me. Tomorrow is the deadline of my assignment then I don't know how to solve that
Rima
i have a question.
Abdul
how do you find the real and complex roots of a polynomial?
Abdul
@abdul with delta maybe which is b(square)-4ac=result then the 1st root -b-radical delta over 2a and the 2nd root -b+radical delta over 2a. I am not sure if this was your question but check it up
Nare
This is the actual question: Find all roots(real and complex) of the polynomial f(x)=6x^3 + x^2 - 4x + 1
Abdul
@Nare please let me know if you can solve it.
Abdul
I have a question
juweeriya
hello guys I'm new here? will you happy with me
mustapha
The average annual population increase of a pack of wolves is 25.
how do you find the period of a sine graph
Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period
Am
if not then how would I find it from a graph
Imani
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
Am
you could also do it with two consecutive minimum points or x-intercepts
Am
I will try that thank u
Imani
Case of Equilateral Hyperbola
ok
Zander
ok
Shella
f(x)=4x+2, find f(3)
Benetta
f(3)=4(3)+2 f(3)=14
lamoussa
14
Vedant
pre calc teacher: "Plug in Plug in...smell's good" f(x)=14
Devante
8x=40
Chris
Explain why log a x is not defined for a < 0
the sum of any two linear polynomial is what
divide simplify each answer 3/2÷5/4
divide simplify each answer 25/3÷5/12
Momo
how can are find the domain and range of a relations
the range is twice of the natural number which is the domain
Morolake
A cell phone company offers two plans for minutes. Plan A: $15 per month and$2 for every 300 texts. Plan B: $25 per month and$0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
6000
Robert
more than 6000
Robert
For Plan A to reach $27/month to surpass Plan B's$26.50 monthly payment, you'll need 3,000 texts which will cost an additional \$10.00. So, for the amount of texts you need to send would need to range between 1-100 texts for the 100th increment, times that by 3 for the additional amount of texts...
Gilbert
...for one text payment for 300 for Plan A. So, that means Plan A; in my opinion is for people with text messaging abilities that their fingers burn the monitor for the cell phone. While Plan B would be for loners that doesn't need their fingers to due the talking; but those texts mean more then...
Gilbert