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Section exercises

Verbal

What is an n th partial sum?

An n th partial sum is the sum of the first n terms of a sequence.

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

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What is a geometric series?

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

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How is finding the sum of an infinite geometric series different from finding the n th partial sum?

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What is an annuity?

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

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Algebraic

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

The sum of terms m 2 + 3 m from m = 1 to m = 5

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

n = 0 4 5 n

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

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The sum that results from adding the number 4 five times

k = 1 5 4

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

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

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10 + 18 + 26 + + 162

k = 1 20 8 k + 2

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1 2 + 1 + 3 2 + 2 + + 4

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

3 2 + 2 + 5 2 + 3 + 7 2

S 5 = 5 ( 3 2 + 7 2 ) 2

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3.2 + 3.4 + 3.6 + + 5.6

S 13 = 13 ( 3.2 + 5.6 ) 2

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

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

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8 + 4 + 2 + + 0.125

k = 1 7 8 0.5 k 1

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1 6 + 1 12 1 24 + + 1 768

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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 + 1 3 + 1 9

S 5 = 9 ( 1 ( 1 3 ) 5 ) 1 1 3 = 121 9 13.44

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n = 1 9 5 2 n 1

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a = 1 11 64 0.2 a 1

S 11 = 64 ( 1 0.2 11 ) 1 0.2 = 781 , 249 , 984 9 , 765 , 625 80

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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.

2 + 1.6 + 1.28 + 1.024 + ...

The series is defined. S = 2 1 0.8

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k = 1 ( 1 2 ) k 1

The series is defined. S = 1 1 ( 1 2 )

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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.

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

Graph of Javier's deposits where the x-axis is the months of the year and the y-axis is the sum of deposits.
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For the following exercises, use the geometric series k = 1 ( 1 2 ) k .

Graph the first 7 partial sums of the series.

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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 k = 1 ( 1 2 ) k is a defined infinite geometric series with S = 1 2 1 ( 1 2 ) = 1.

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Numeric

For the following exercises, find the indicated sum.

n = 1 6 n ( n 2 )

49

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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

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6 + 15 2 + 9 + 21 2 + 12 + 27 2 + 15

S 7 = 147 2

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1 + 3 + 7 + ... + 31

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k = 1 11 ( k 2 1 2 )

S 11 = 55 2

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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...

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S 7 for the series 0.4 2 + 10 50...

S 7 = 5208.4

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n = 1 10 2 ( 1 2 ) n 1

S 10 = 1023 256

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

1 1 4 1 16 1 64 ...

S = 4 3

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k = 1 3 ( 1 4 ) k 1

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n = 1 4.6 0.5 n 1

S = 9.2

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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: $ 50 ; total deposits: 60 ; interest rate: 5 % , compounded monthly

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

$3,705.42

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Deposit amount: $ 450 ; total deposits: 60 ; interest rate: 4.5 % , compounded quarterly

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Deposit amount: $ 100 ; total deposits: 120 ; interest rate: 10 % , compounded semi-annually

$695,823.97

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Extensions

The sum of terms 50 k 2 from k = x through 7 is 115. What is x ?

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Write an explicit formula for a k such that k = 0 6 a k = 189. Assume this is an arithmetic series.

a k = 30 k

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Find the smallest value of n such that k = 1 n ( 3 k 5 ) > 100.

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How many terms must be added before the series 1 3 5 7 ....   has a sum less than 75 ?

9 terms

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Write 0. 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. 65 ¯ to a fraction.

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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 = 4 5

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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?

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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

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Real-world applications

Keisha devised a week-long study plan to prepare for finals. On the first day, she plans to study for 1 hour, and each successive day she will increase her study time by 30 minutes. How many hours will Keisha have studied after one week?

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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

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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?

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A pendulum travels a distance of 3 feet on its first swing. On each successive swing, it travels 3 4 the distance of the previous swing. What is the total distance traveled by the pendulum when it stops swinging?

12 feet

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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?

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Questions & Answers

bsc F. y algebra and trigonometry pepper 2
Aditi Reply
given that x= 3/5 find sin 3x
Adamu Reply
4
DB
remove any signs and collect terms of -2(8a-3b-c)
Joeval Reply
-16a+6b+2c
Will
is that a real answer
Joeval
(x2-2x+8)-4(x2-3x+5)
Ayush Reply
sorry
Miranda
x²-2x+9-4x²+12x-20 -3x²+10x+11
Miranda
x²-2x+9-4x²+12x-20 -3x²+10x+11
Miranda
(X2-2X+8)-4(X2-3X+5)=0 ?
master
The anwser is imaginary number if you want to know The anwser of the expression you must arrange The expression and use quadratic formula To find the answer
master
The anwser is imaginary number if you want to know The anwser of the expression you must arrange The expression and use quadratic formula To find the answer
master
Y
master
X2-2X+8-4X2+12X-20=0 (X2-4X2)+(-2X+12X)+(-20+8)= 0 -3X2+10X-12=0 3X2-10X+12=0 Use quadratic formula To find the answer answer (5±Root11i)/3
master
Soo sorry (5±Root11* i)/3
master
x2-2x+8-4x2+12x-20 x2-4x2-2x+12x+8-20 -3x2+10x-12 now you can find the answer using quadratic
Mukhtar
explain and give four example of hyperbolic function
Lukman Reply
What is the correct rational algebraic expression of the given "a fraction whose denominator is 10 more than the numerator y?
Racelle Reply
y/y+10
Mr
Find nth derivative of eax sin (bx + c).
Anurag Reply
Find area common to the parabola y2 = 4ax and x2 = 4ay.
Anurag
A rectangular garden is 25ft wide. if its area is 1125ft, what is the length of the garden
Jhovie Reply
to find the length I divide the area by the wide wich means 1125ft/25ft=45
Miranda
thanks
Jhovie
What do you call a relation where each element in the domain is related to only one value in the range by some rules?
Charmaine Reply
A banana.
Yaona
given 4cot thither +3=0and 0°<thither <180° use a sketch to determine the value of the following a)cos thither
Snalo Reply
what are you up to?
Mark Reply
nothing up todat yet
Miranda
hi
jai
hello
jai
Miranda Drice
jai
aap konsi country se ho
jai
which language is that
Miranda
I am living in india
jai
good
Miranda
what is the formula for calculating algebraic
Propessor Reply
I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
Miranda
state and prove Cayley hamilton therom
sita Reply
hello
Propessor
hi
Miranda
the Cayley hamilton Theorem state if A is a square matrix and if f(x) is its characterics polynomial then f(x)=0 in another ways evey square matrix is a root of its chatacteristics polynomial.
Miranda
hi
jai
hi Miranda
jai
thanks
Propessor
welcome
jai
What is algebra
Pearl Reply
algebra is a branch of the mathematics to calculate expressions follow.
Miranda
Miranda Drice would you mind teaching me mathematics? I think you are really good at math. I'm not good at it. In fact I hate it. 😅😅😅
Jeffrey
lolll who told you I'm good at it
Miranda
something seems to wispher me to my ear that u are good at it. lol
Jeffrey
lolllll if you say so
Miranda
but seriously, Im really bad at math. And I hate it. But you see, I downloaded this app two months ago hoping to master it.
Jeffrey
which grade are you in though
Miranda
oh woww I understand
Miranda
haha. already finished college
Jeffrey
how about you? what grade are you now?
Jeffrey
I'm going to 11grade
Miranda
how come you finished in college and you don't like math though
Miranda
gotta practice, holmie
Steve
if you never use it you won't be able to appreciate it
Steve
I don't know why. But Im trying to like it.
Jeffrey
yes steve. you're right
Jeffrey
so you better
Miranda
what is the solution of the given equation?
Nelson Reply
which equation
Miranda
I dont know. lol
Jeffrey
please where is the equation
Miranda
ask nelson. lol
Jeffrey

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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