<< Chapter < Page Chapter >> Page >

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.

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

What is a geometric series?

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

What is an annuity?

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

n = 0 4 5 n

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

The sum that results from adding the number 4 five times

k = 1 5 4

Got questions? Get instant answers now!

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

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

Got questions? Get instant answers now!

10 + 18 + 26 + + 162

k = 1 20 8 k + 2

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

3.2 + 3.4 + 3.6 + + 5.6

S 13 = 13 ( 3.2 + 5.6 ) 2

Got questions? Get instant answers now!

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

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

Got questions? Get instant answers now!

8 + 4 + 2 + + 0.125

k = 1 7 8 0.5 k 1

Got questions? Get instant answers now!

1 6 + 1 12 1 24 + + 1 768

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

n = 1 9 5 2 n 1

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

k = 1 ( 1 2 ) k 1

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

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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.
Got questions? Get instant answers now!

For the following exercises, use the geometric series k = 1 ( 1 2 ) k .

Graph the first 7 partial sums of the series.

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

Numeric

For the following exercises, find the indicated sum.

n = 1 6 n ( n 2 )

49

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

S 7 = 147 2

Got questions? Get instant answers now!

1 + 3 + 7 + ... + 31

Got questions? Get instant answers now!

k = 1 11 ( k 2 1 2 )

S 11 = 55 2

Got questions? Get instant answers now!

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...

Got questions? Get instant answers now!

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

S 7 = 5208.4

Got questions? Get instant answers now!

n = 1 10 2 ( 1 2 ) n 1

S 10 = 1023 256

Got questions? Get instant answers now!

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

1 1 4 1 16 1 64 ...

S = 4 3

Got questions? Get instant answers now!

k = 1 3 ( 1 4 ) k 1

Got questions? Get instant answers now!

n = 1 4.6 0.5 n 1

S = 9.2

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

$3,705.42

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

$695,823.97

Got questions? Get instant answers now!

Extensions

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

9 terms

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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?

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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?

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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?

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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?

Got questions? Get instant answers now!

Questions & Answers

For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
Shakeena Reply
by how many trees did forest "A" have a greater number?
Shakeena
32.243
Kenard
how solve standard form of polar
Rhudy Reply
what is a complex number used for?
Drew Reply
It's just like any other number. The important thing to know is that they exist and can be used in computations like any number.
Steve
I would like to add that they are used in AC signal analysis for one thing
Scott
Good call Scott. Also radar signals I believe.
Steve
Is there any rule we can use to get the nth term ?
Anwar Reply
how do you get the (1.4427)^t in the carp problem?
Gabrielle Reply
A hedge is contrusted to be in the shape of hyperbola near a fountain at the center of yard.the hedge will follow the asymptotes y=x and y=-x and closest distance near the distance to the centre fountain at 5 yards find the eqution of the hyperbola
ayesha Reply
A doctor prescribes 125 milligrams of a therapeutic drug that decays by about 30% each hour. To the nearest hour, what is the half-life of the drug?
Sandra Reply
Find the domain of the function in interval or inequality notation f(x)=4-9x+3x^2
prince Reply
hello
Jessica Reply
Outside temperatures over the course of a day can be modeled as a sinusoidal function. Suppose the high temperature of ?105°F??105°F? occurs at 5PM and the average temperature for the day is ?85°F.??85°F.? Find the temperature, to the nearest degree, at 9AM.
Karlee Reply
if you have the amplitude and the period and the phase shift ho would you know where to start and where to end?
Jean Reply
rotation by 80 of (x^2/9)-(y^2/16)=1
Garrett Reply
thanks the domain is good but a i would like to get some other examples of how to find the range of a function
bashiir Reply
what is the standard form if the focus is at (0,2) ?
Lorejean Reply
a²=4
Roy Reply

Get the best Precalculus course in your pocket!





Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Precalculus' conversation and receive update notifications?

Ask