<< 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

A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
Kaitlyn Reply
The sequence is {1,-1,1-1.....} has
amit Reply
circular region of radious
Kainat Reply
how can we solve this problem
Joel Reply
Sin(A+B) = sinBcosA+cosBsinA
Eseka Reply
Prove it
Eseka
Please prove it
Eseka
hi
Joel
June needs 45 gallons of punch. 2 different coolers. Bigger cooler is 5 times as large as smaller cooler. How many gallons in each cooler?
Arleathia Reply
7.5 and 37.5
Nando
find the sum of 28th term of the AP 3+10+17+---------
Prince Reply
I think you should say "28 terms" instead of "28th term"
Vedant
the 28th term is 175
Nando
192
Kenneth
if sequence sn is a such that sn>0 for all n and lim sn=0than prove that lim (s1 s2............ sn) ke hole power n =n
SANDESH Reply
write down the polynomial function with root 1/3,2,-3 with solution
Gift Reply
if A and B are subspaces of V prove that (A+B)/B=A/(A-B)
Pream Reply
write down the value of each of the following in surd form a)cos(-65°) b)sin(-180°)c)tan(225°)d)tan(135°)
Oroke Reply
Prove that (sinA/1-cosA - 1-cosA/sinA) (cosA/1-sinA - 1-sinA/cosA) = 4
kiruba Reply
what is the answer to dividing negative index
Morosi Reply
In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
Shivam Reply
give me the waec 2019 questions
Aaron Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

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

Ask