Section 25–3
◆
709
Geometric Progressions
Harmonic Progressions
25. Find the fourth term of the harmonic progression
3 3 3
, , , . . .
5 8 11
26. Find the fifth term of the harmonic progression
4 4 4
, , , . . .
19 15 11
Harmonic Means
7
27. Insert two harmonic means between 79 and .
15
6
28. Insert three harmonic means between and 65 .
21
Applications
29. Loan Repayment: A person agrees to repay a loan of $10,000 with an annual payment of
$1,000 plus 8% of the unpaid balance.
(a) Show that the interest payments alone form the AP: $800, $720, $640, . . . .
(b) Find the total amount of interest paid.
30. Simple Interest: A person deposits $50 in a bank on the first day of each month, at the same
time withdrawing all interest earned on the money already in the account.
(a) If the rate is 1% per month, computed monthly, write an AP whose terms are the
amounts withdrawn each month.
(b) How much interest will have been earned in the 36 months following the first deposit?
31. Straight-Line Depreciation: A certain milling machine has an initial value of $150,000 and
a scrap value of $10,000 twenty years later. Assuming that the machine depreciates the
same amount each year, find its value after 8 years.
32. Salary or Price Increase: A person is hired at a salary of $40,000 and receives a raise of
$2,500 at the end of each year. Find the total amount earned during 10 years.
33. Freely Falling Body: A freely falling body falls gⲐ2 metres during the first second, 3gⲐ2 m
during the next second, 5gⲐ2 m during the third second, and so on, where g ⬵ 9.807 mⲐs2.
Find the total distance the body falls during the first 10 s.
34. Using the information of problem 33, show that the total distance s fallen in t seconds is
s 12 gt 2.
25–3 Geometric Progressions
Recursion Formula
A geometric sequence or geometric progression (GP) is one in which each term after the first is
formed by multiplying the preceding term by a factor r, called the common ratio. Thus if an is
any term of a GP, the recursion relation is as follows:
GP:
Recursion
Formula
an ran1
240
Each term of a GP after the first equals the product of the preceding term and the common ratio.
To find the amount of
depreciation for each year, divide
the total depreciation (initial
value – scrap value) by the
number of years of depreciation.
710
Chapter 25
◆
Sequences, Series, and the Binomial Theorem
◆◆◆
Example 21: Some geometric progressions, with their common ratios given, are as follows:
(a) 2, 4, 8, 16, . . .
(r 2)
1
(b) 27, 9, 3, 1, 3 , . . .
(r 13 )
(c) 1, 3, 9, 27, . . .
(r 3)
◆◆◆
General Term
For a GP whose first term is a and whose common ratio is r, the terms are
a, ar, ar2, ar3, ar4, . . .
We see that each term after the first is the product of the first term and a power of r, where the
power of r is one less than the number n of the term. So the nth term an is given by the following equation:
GP:
General
Term
an arn1
241
The nth term of a GP is found by multiplying the first term by the n 1 power of the common ratio.
◆◆◆
Example 22: Find the sixth term of a GP with first term 5 and common ratio 4.
Solution: We substitute into Eq. 241 for the general term of a GP, with a 5, n 6, and r 4.
a6 5(45) 5(1024) 5120
◆◆◆
GP: Sum of n Terms
We find a formula for the sum sn of the first n terms of a GP (also called the sum of n terms of
a geometric series) by adding the terms of the GP.
sn a ar ar2 ar3 arn2 arn1
(1)
Multiplying each term in Eq. (1) by r gives
rsn ar ar2 ar3 arn1 arn
(2)
Subtracting (2) from (1) term by term, we get
(1 r)sn a arn
Dividing both sides by (1 r) gives us the following formula:
GP: Sum of
n Terms
◆◆◆
a(1 rn)
sn 1r
242
Example 23: Find the sum of the first six terms of the GP in Example 22.
Solution: We substitute into Eq. 242 using a 5, n 6, and r 4.
a (1 rn) 5 (1 46) 5 (4095)
sn 6825
1r
14
3
◆◆◆
We can get another equation for the sum of a GP in terms of the nth term an. We substitute
into Eq. 242, using arn r (arn1) ran, as follows:
Section 25–3
◆
711
Geometric Progressions
a ran
sn 1r
GP: Sum of
n Terms
243
◆◆◆
Example 24: Repeat Example 23, given that the sixth term (found in Example 22) is
a6 5120.
Solution: Substitution, with a 5, r 4, and a6 5120, yields
a ran 5 4 (5120)
sn 6825
1r
14
◆◆◆
as before.
Geometric Means
As with the AP, the intermediate terms between any two terms are called means. A single number inserted between two numbers is called the geometric mean between those numbers.
◆◆◆
Example 25: Insert a geometric mean b between two numbers a and c.
Solution: Our GP is a, b, c. The common ratio r is then
b c
r a b
2
from which b ac, which is rewritten as follows:
Geometric
Mean
b 兹 ac
59
The geometric mean, or mean proportional, between two numbers is equal to the square
◆◆◆
root of their product.
◆◆◆
Example 26: Find the geometric mean between 3 and 48.
Solution: Letting a 3 and c 48 gives us
b 兹 3 (48) 12
Our GP is then
3, 12, 48
or
3, ⴚ12, 48
◆◆◆
Note that we get two solutions.
To insert several geometric means between two numbers, we first find the common ratio.
3
Example 27: Insert four geometric means between 2 and 15 .
16
3
Solution: Here a 2, a6 15
, and n 6. Then
16
◆◆◆
a6 ar5
3
15 2r5
16
This is not new. We studied the
mean proportional in Chapter 19.
712
Chapter 25
◆
Sequences, Series, and the Binomial Theorem
243
r5 32
3
r 2
Having r, we can write the terms of the GP. They are
2,
Exercise 3
1.
2.
3.
4.
5.
6.
7.
8.
◆
3,
1
4 ,
2
3
6 ,
4
1
10 ,
8
3
15 16
◆◆◆
Geometric Progressions
Find the fifth term of a GP with first term 5 and common ratio 2.
Find the fourth term of a GP with first term 7 and common ratio 4.
Find the sixth term of a GP with first term 3 and common ratio 5.
Find the fifth term of a GP with first term 4 and common ratio 2.
Find the sum of the first ten terms of the GP in problem 1.
Find the sum of the first nine terms of the GP in problem 2.
Find the sum of the first eight terms of the GP in problem 3.
Find the sum of the first five terms of the GP in problem 4.
Geometric Means
9.
10.
11.
12.
13.
14.
15.
16.
Insert a geometric mean between 5 and 45.
Insert a geometric mean between 7 and 112.
Insert a geometric mean between 10 and 90.
Insert a geometric mean between 21 and 84.
Insert two geometric means between 8 and 216.
Insert two geometric means between 9 and 243.
Insert three geometric means between 5 and 1280.
Insert three geometric means between 144 and 9.
Applications
17. Exponential Growth: Using the equation for exponential growth:
y aent
199
with a 1 and n 0.5, compute values of y for t 0, 1, 2, . . . , 10. Show that while the
values of t form an AP, the values of y form a GP. Find the common ratio.
18. Exponential Decay: Repeat problem 17 with the formula for exponential decay:
y aent
One of the most famous and
controversial references to
arithmetic and geometric
progressions was made by
Thomas Malthus in 1798.
He wrote: “Population, when
unchecked, increases in a
geometrical ratio,and subsistence
for man in an arithmetical ratio.”
201
19. Cooling: A certain iron casting is at 950 C and cools so that its temperature at each minute
is 10% less than its temperature the preceding minute. Find its temperature after 1 h.
20. Light through an Absorbing Medium: Sunlight passes through a glass filter. Each millimetre of glass absorbs 20% of the light passing through it. What percentage of the original
sunlight will remain after passing through 5.0 mm of the glass?
21. Radioactive Decay: A certain radioactive material decays so that after each year the radioactivity is 8% less than at the start of that year. How many years will it take for its radioactivity to be 50% of its original value?
Section 25–4
◆
Infinite Geometric Progressions
22. Pendulum: Each swing of a certain pendulum is 85.0% as long as the one before. If the first
swing is 12.0 cm, find the entire distance travelled in eight swings.
23. Bouncing Ball: A ball dropped from a height of 3.00 m rebounds to half its height on each
bounce. Find the total distance travelled when it hits the ground for the fifth time.
24. Population Growth: Each day the size of a certain colony of bacteria is 25% larger than on
the preceding day. If the original size of the colony was 10 000 bacteria, find its size after
5 days.
25. Ancestry: A person has two parents, and each parent has two parents, and so on. We can
write a GP for the number of ancestors as 2, 4, 8, . . . . Find the total number of ancestors
in five generations, starting with the parents’ generation.
26. Musical Scale: The frequency of the “A” note above middle C is, by international agreement, equal to 440 Hz. A note one octave higher is at twice that frequency, or 880 Hz. The
octave is subdivided into 12 half-tone intervals, where each half-tone is higher than the one
preceding by a factor equal to the twelfth root of 2. Write a GP showing the frequency of
each half-tone, from 440 to 880 Hz. Work to two decimal places.
27. Chemical Reactions: Increased temperature usually causes chemicals to react faster. If a
certain reaction proceeds 15% faster for each 10 C increase in temperature, by what factor
is the reaction speed increased when the temperature rises by 50 C?
28. Mixtures: A radiator contains 30% antifreeze and 70% water. One-fourth of the mixture is
removed and replaced by pure water. If this procedure is repeated three more times, find
the percent antifreeze in the final mixture.
29. Energy Consumption: If energy consumption in Canada is 7.00% higher each year, by
what factor will the energy consumption have increased after 10.0 years?
30. Atmospheric Pressure: The pressures measured at 1-km intervals above sea level form a
GP, with each value smaller than the preceding by a factor of 0.882. If the pressure at sea
level is 101.1 kPa, find the pressure at an altitude of 8 km.
31. Compound Interest: A person deposits $10,000 in a bank giving 6% interest, compounded
annually. Find to the nearest dollar the value of the deposit after 50 years.
32. Inflation: The price of a certain house, now $126,000, is expected to increase by 5% each
year. Write a GP whose terms are the value of the house at the end of each year, and find
the value of the house after 5 years.
33. Depreciation: When calculating depreciation by the declining-balance method, a taxpayer
claims as a deduction a fixed percentage of the book value of an asset each year. The new
book value is then the last book value less the amount of the depreciation. Thus for a
machine having an initial book value of $100,000 and a depreciation rate of 40%, the first
year’s depreciation is 40% of $100,000, or $40,000, and the new book value is $100,000 $40,000 $60,000. Thus the book values for each year form the following GP:
$100,000, $60,000, $36,000, . . .
Find the book value after 5 years.
25–4 Infinite Geometric Progressions
Sum of an Infinite Geometric Progression
Before we derive a formula for the sum of an infinite geometric progression, let us explore the
idea graphically and numerically.
We have already determined that the sum of n terms of any geometric progression with a
first term a and a common ratio r is
a (1 r n )
sn (Eq. 242)
1r
Thus a graph of sn versus n should tell us about the sum changes as n increases.
713
This is called the equally
tempered scale and is usually
attributed to Johann Sebastian
Bach (1685–1750).