Applied Calculus

Deborah Hughes-Hallett, Patti Frazer Lock, Andrew M. Gleason

Chapter 10

Geometric Series - all with Video Answers

Educators

Section 1

Geometric Series

Problem 1

Find the sum of the following series in two ways: by adding terms and by using the geometric series formula.
$$3+3 \cdot 2+3 \cdot 2^{2}$$

Vikash Ranjan

Vikash Ranjan

Numerade Educator

Problem 2

Find the sum of the following series in two ways: by adding terms and by using the geometric series formula.
$$50+50(0.9)+50(0.9)^{2}+50(0.9)^{3}$$

Vikash Ranjan

Numerade Educator

Problem 3

Decide which of the following are geometric series. For those which are, give the first term and the ratio between successive terms. For those which are not, explain why not.
$$5-10+20-40+80-\dots$$

Vikash Ranjan

Numerade Educator

Problem 4

Decide which of the following are geometric series. For those which are, give the first term and the ratio between successive terms. For those which are not, explain why not.
$$1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\cdots$$

Vikash Ranjan

Numerade Educator

Problem 5

Decide which of the following are geometric series. For those which are, give the first term and the ratio between successive terms. For those which are not, explain why not.
$$2+1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots$$

Vikash Ranjan

Numerade Educator

Problem 6

Decide which of the following are geometric series. For those which are, give the first term and the ratio between successive terms. For those which are not, explain why not.
$$1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\frac{1}{16}+\cdots$$

Vikash Ranjan

Numerade Educator

Problem 7

Decide which of the following are geometric series. For those which are, give the first term and the ratio between successive terms. For those which are not, explain why not.
$$1+2 z+(2 z)^{2}+(2 z)^{3}+\cdots$$

Vikash Ranjan

Numerade Educator

Problem 8

Decide which of the following are geometric series. For those which are, give the first term and the ratio between successive terms. For those which are not, explain why not.
$$1+x+2 x^{2}+3 x^{3}+4 x^{4}+\cdots$$

Vikash Ranjan

Numerade Educator

Problem 9

Decide which of the following are geometric series. For those which are, give the first term and the ratio between successive terms. For those which are not, explain why not.
$$y^{2}+y^{3}+y^{4}+y^{5}+\cdots$$

Vikash Ranjan

Numerade Educator

Problem 10

Find the sum, if it exists.
$$5+5 \cdot 3+5 \cdot 3^{2}+\dots+5 \cdot 3^{12}$$

Vikash Ranjan

Numerade Educator

Problem 11

Find the sum, if it exists.
$$100+100(0.85)+100(0.85)^{2}+\cdots+100(0.85)^{10}$$

Vikash Ranjan

Numerade Educator

Problem 12

Find the sum, if it exists.
$$1000+1000(1.05)+1000(1.05)^{2}+\cdots$$

Bobby Barnes

University of North Texas

Problem 13

Find the sum, if it exists.
$$75+75(0.22)+75(0.22)^{2}+\cdots$$

Vikash Ranjan

Numerade Educator

Problem 14

Find the sum, if it exists.
$$20+20(1.45)+20(1.45)^{2}+\cdots+20(1.45)^{14}$$

Vikash Ranjan

Numerade Educator

Problem 15

Find the sum, if it exists.
$$500(0.4)+500(0.4)^{2}+500(0.4)^{3}+\cdots$$

Vikash Ranjan

Numerade Educator

Problem 16

Find the sum, if it exists.
$$31500+6300+1260+252+\cdots$$

Vikash Ranjan

Numerade Educator

Problem 17

Find the sum, if it exists.
$$3+\frac{3}{2}+\frac{3}{4}+\frac{3}{8}+\dots+\frac{3}{2^{10}}$$

Vikash Ranjan

Numerade Educator

Problem 18

Find the sum, if it exists.
$$1000+1500+2250+3375+5062.5+\dots$$

Vikash Ranjan

Numerade Educator

Problem 19

Find the sum, if it exists.
$$200+100+50+25+12.5+\cdots$$

Vikash Ranjan

Numerade Educator

Problem 20

Find the sum, if it exists.
$$65+\frac{65}{1.02}+\frac{65}{(1.02)^{2}}+\dots+\frac{65}{(1.02)^{18}}$$

Vikash Ranjan

Numerade Educator

Problem 21

Find the sum, if it exists.
$$-2+1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\frac{1}{16}-\cdots$$

Vikash Ranjan

Numerade Educator

Problem 22

In Example 3(b) on page $467,$ we found partial sums for the geometric series with $a=250$ and $r=1.2 .$ Find the partial sums $S_{n}$ for $n=5,10,15,20 .$ As $n$ gets larger, do the partial sums appear to grow without bound, as expected if $r>1 ?$

Vikash Ranjan

Numerade Educator

Problem 23

In Example 3(a), we found partial sums of the geometric series with $a=10$ and $r=0.75$ and showed that the sum of this series is 40. Find the partial sums $S_{n}$ for $n=5,10,15,20 .$ As $n$ gets larger, do the partial sums appear to be approaching 40?

Vikash Ranjan

Numerade Educator

Problem 24

A repeating decimal can always be expressed as a fraction. This problem shows how writing a repeating decimal as a geometric series enables you to find the fraction.
(a) Write the repeating decimal $0.232323 \ldots$ as a geometric series using the fact that $0.232323 \ldots=$ $0.23+0.0023+0.000023+\cdots.$
(b) Use the formula for the sum of a geometric series to show that $0.232323 \ldots=23 / 99.$

Vikash Ranjan

Numerade Educator

Problem 25

Every month, $\$ 500$ is deposited into an account earning $0.1 \%$ interest a month, compounded monthly.
(a) How much is in the account right after the $6^{\text {th }}$ deposit? Right before the $6^{\text {th }}$ deposit?
(b) How much is in the account right after the $12^{\text {th }}$ deposit? Right before the $12^{\text {th }}$ deposit?

Vikash Ranjan

Numerade Educator

Problem 26

Each year, a family deposits $\$ 5000$ into an account paying $1.25 \%$ interest per year, compounded annually. How much is in the account right after the $25^{\text {th }}$ deposit?

Nick Johnson

Numerade Educator

Problem 27

A smoker inhales 0.4 mg of nicotine from a cigarette. After one hour, 71% of the nicotine remains in the body. If a person smokes one cigarette every hour beginning at 7 am, how much nicotine is in the body right after the 11 pm cigarette?

Vikash Ranjan

Numerade Educator

Problem 28

Each morning, a patient receives a 25 mg injection of an anti-inflammatory drug, and 40% of the drug remains in the body after 24 hours. Find the quantity in the body:
(a) Right after the $3^{\text {rd }}$ injection.
(b) Right after the $6^{\text {th }}$ injection.
(c) In the long run, right after an injection.

Vikash Ranjan

Numerade Educator

Problem 29

In Example 4 on page 468, we saw that if 50 mg of quinine is given every 24 hours, the long-run quantity of quinine in the body is about 65 mg right after a dose and about 15 mg right before a dose. The concentration of quinine in the body is measured in milligrams of quinine per kilogram of body weight. To be effective, the average concentration of quinine in the body must be at least 0.4 mg/kg. Concentrations above 3.0 mg/kg are not safe.
(a) Estimate the average quantity of quinine in the body in the long run by averaging the long-run quantities of quinine in the body right after a dose and right before a dose.
(b) Find the average concentration for a person weighing 70 kilograms. Is this treatment safe and effective for such a person?
(c) For what range of weights would this treatment produce a long-run average concentration that is
(i) Too low?
(ii) Unsafe?

Vikash Ranjan

Numerade Educator

Problem 30

A ball is dropped from a height of 10 feet and bounces. Each bounce is $\frac{3}{4}$ of the height of the bounce before. Thus, after the ball hits the floor for the first time, the ball rises to a height of $10\left(\frac{3}{4}\right)=7.5$ feet, and after it hits the floor for the second time, it rises to a height of $7.5\left(\frac{3}{4}\right)=10\left(\frac{3}{4}\right)^{2}=5.625$ feet.
(a) Find an expression for the height to which the ball rises after it hits the floor for the $n^{\text {th }}$ time.
(b) Find an expression for the total vertical distance the ball has traveled when it hits the floor for the first, second, third, and fourth times.
(c) Find an expression for the total vertical distance the ball has traveled when it hits the floor for the $n^{\mathrm{th}}$ time. Express your answer in closed form.

Vikash Ranjan

Numerade Educator