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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

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
what is titration
John Reply
what is physics
Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
David Reply
what is viscosity?
David
what is inorganic
emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
what's motion
Maurice Reply
what are the types of wave
Maurice
answer
Magreth
progressive wave
Magreth
hello friend how are you
Muhammad Reply
fine, how about you?
Mohammed
hi
Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
yasuo Reply
Who can show me the full solution in this problem?
Reofrir Reply
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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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