Tan S.T. Finite Mathematics for the Managerial, Life, and Social Sciences
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APPLIED EXAMPLE 4
Company Sales Madison Electric Company
had sales of $200,000 in its first year of operation. If the sales increased
by $30,000 per year thereafter, find Madison’s sales in the fifth year and its total
sales over the first 5 years of operation.
Solution
Madison’s yearly sales follow an arithmetic progression, with the first
term given by a 200,000 and the common difference given by d 30,000. The
sales in the fifth year are found by using Equation (16) with n 5. Thus,
a
5
200,000 (5 1)30,000 320,000
or $320,000.
Madison’s total sales over the first 5 years of operation are found by using
Equation (19) with n 5. Thus,
or $1,300,000.
Geometric Progressions
A geometric progression is a sequence of numbers in which each term after the first
is obtained by multiplying the preceding term by a constant r. The constant r is called
the common ratio.
A geometric progression is completely determined if the first term and the com-
mon ratio are known. Thus, if
a
1
, a
2
, a
3
, . . . , a
n
, . . .
is a geometric progression with the first term given by a and common ratio given by
r, then by definition we have
This gives the following:
EXAMPLE 5
Find the eighth term of a geometric progression whose first five
terms are 162, 54, 18, 6, and 2.
Solution
The common ratio is found by taking the ratio of any term other than the
first to the preceding term. Taking the ratio of the fourth term to the third term, for
nth Term of a Geometric Progression
The nth term of a geometric progression with first term a and common ratio r is
given by
a
n
ar
n1
(20)
a
n
a
n 1
r ar
n 1
o
a
4
a
3
r ar
3
a
3
a
2
r ar
2
a
2
a
1
r ar
a
1
a
1,300,000
S
5
5
2
321200,0002 15 1230,000 4
302 5 MATHEMATICS OF FINANCE
87533_05_ch5_p257-312 1/30/08 9:55 AM Page 302
example, gives . To find the eighth term of the geometric progression, use
Formula (20) with a 162, , and n 8, obtaining
EXAMPLE 6
Find the tenth term of a geometric progression with positive terms
and third term equal to 16 and seventh term equal to 1.
Solution
Using Equation (20) with n 3 and n 7, respectively, yields
Dividing by a
3
gives
from which we obtain . Substituting this value of r into the expres-
sion for a
3
, we obtain
Finally, using (20) once again with a 64, , and n 10 gives
To find the sum of the first n terms of a geometric progression with the first term
a
1
a and common ratio r, denote the required sum by S
n
. Then,
S
n
a ar ar
2
ar
n2
ar
n1
(21)
Upon multiplying (21) by r, we obtain
rS
n
ar ar
2
ar
3
ar
n1
ar
n
(22)
Subtracting (22) from (21) gives
If r 1, we may divide both sides of the last equation by (1 r), obtaining
If r 1, then (21) gives
S
n
a a a a
n
terms
na
Thus,
S
n
•
a11 r
n
2
1 r
if r 1
na if r 1
S
n
a11 r
n
2
1 r
11 r2S
n
a11 r
n
2
S
n
rS
n
a ar
n
a
10
64 a
1
2
b
9
1
8
r
1
2
a a
1
2
b
2
16
or
a 64
r
4
1
16
, or r
1
2
ar
6
ar
2
1
16
a
7
a
7
ar
6
1
a
3
ar
2
16
2
27
a
8
162 a
1
3
b
7
r
1
3
r
6
18
1
3
5.4 ARITHMETIC AND GEOMETRIC PROGRESSIONS 303
87533_05_ch5_p257-312 1/30/08 9:55 AM Page 303
EXAMPLE 7
Find the sum of the first six terms of the following geometric
progression:
3, 6, 12, 24, . . .
Solution
Here, a 3, 2, and n 6, so Formula (23) gives
APPLIED EXAMPLE 8
Company Sales Michaelson Land Develop-
ment Company had sales of $1 million in its first year of operation. If
sales increased by 10% per year thereafter, find Michaelson’s sales in the fifth
year and its total sales over the first 5 years of operation.
Solution
Michaelson’s yearly sales follow a geometric progression, with the
first term given by a 1,000,000 and the common ratio given by r 1.1. The
sales in the fifth year are found by using Formula (20) with n 5. Thus,
a
5
1,000,000(1.1)
4
1,464,100
or $1,464,100.
Michaelson’s total sales over the first 5 years of operation are found by using
Equation (23) with n 5. Thus,
or $6,105,100.
Double Declining-Balance Method of Depreciation
In Section 1.3, we discussed the straight-line, or linear, method of depreciating an
asset. Linear depreciation assumes that the asset depreciates at a constant rate. For cer-
tain assets (such as machines) whose market values drop rapidly in the early years of
usage and thereafter less rapidly, another method of depreciation called the double
declining-balance method is often used. In practice, a business firm normally
employs the double declining-balance method for depreciating such assets for a cer-
tain number of years and then switches over to the linear method.
To derive an expression for the book value of an asset being depreciated by
the double declining-balance method, let C (in dollars) denote the original cost of
the asset and let the asset be depreciated over N years. Using this method, the
6,105,100
S
5
1,000,00031 11.12
5
4
1 1.1
S
6
311 2
6
2
1 2
189
r
6
3
304 5 MATHEMATICS OF FINANCE
Sum of Terms in a Geometric Progression
The sum of the first n terms of a geometric progression with first term a and
common ratio r is given by
(23)
S
n
•
a11 r
n
2
1 r
if r 1
na if r 1
87533_05_ch5_p257-312 1/30/08 9:55 AM Page 304
amount depreciated each year is times the value of the asset at the beginning of
that year. Thus, the amount by which the asset is depreciated in its first year of use
is given by , so if V(1) denotes the book value of the asset at the end of the first
year then
Next, if V(2) denotes the book value of the asset at the end of the second year, then a
similar argument leads to
Continuing, we find that if V(n) denotes the book value of the asset at the end of n
years, then the terms C, V(1), V(2), . . . , V(N) form a geometric progression with first
term C and common ratio . Consequently, the nth term, V(n), is given by
(1 n N) (24)
Also, if D(n) denotes the amount by which the asset has been depreciated by the end
of the nth year, then
(25)
APPLIED EXAMPLE 9
Depreciation of Equipment A tractor pur-
chased at a cost of $60,000 is to be depreciated by the double declining-
balance method over 10 years. What is the book value of the tractor at the end
of 5 years? By what amount has the tractor been depreciated by the end of the
fifth year?
Solution
We have C 60,000 and N 10. Thus, using Formula (24) with
n 5 gives the book value of the tractor at the end of 5 years as
or $19,660.80.
The amount by which the tractor has been depreciated by the end of the fifth
year is given by
60,000 19,660.80 40,339.20
or $40,339.20. You may verify the last result by using Equation (25) directly.
60,000 a
4
5
b
5
19,660.80
V152 60,000 a1
2
10
b
5
C c1 a1
2
N
b
n
d
D1n2 C C a1
2
N
b
n
V1n2 C a1
2
N
b
n
11
2
N
2
C a1
2
N
b
2
C a1
2
N
ba1
2
N
b
V122 C a1
2
N
b C a1
2
N
b
2
N
V112 C
2C
N
C a1
2
N
b
2C
N
2
N
5.4 ARITHMETIC AND GEOMETRIC PROGRESSIONS 305
87533_05_ch5_p257-312 1/30/08 9:55 AM Page 305
1. Find the sum of the first five terms of the geometric pro-
gression with first term 24 and common ratio .
2. Office equipment purchased for $75,000 is to be depreci-
ated by the double declining-balance method over 5 yr.
Find the book value at the end of 3 yr.
1
2
306 5 MATHEMATICS OF FINANCE
Exploring with
TECHNOLOGY
A tractor purchased at a cost of $60,000 is to be depreciated over 10 years with
a residual value of $0. Using the double declining-balance method, its value at
the end of n years is V
1
(n) 60,000(0.8)
n
dollars. Using straight-line deprecia-
tion, its value at the end of n years is V
2
(n) 60,000 6000n. Use a graphing
utility to sketch the graphs of V
1
and V
2
in the viewing window [0, 10]
[0, 70,000]. Comment on the relative merits of each method of depreciation.
3. Derive the formula for the future value of an annuity [Equa-
tion (9), Section 5.2].
Solutions to Self-Check Exercises 5.4 can be found on
page 308.
5.4 Exercises
5.4 Self-Check Exercises
5.4 Concept Questions
1. Suppose an arithmetic progression has first term a and
common difference d.
a. What is the formula for the nth term of this progression?
b. What is the formula for the sum of the first n terms of
this progression?
2. Suppose a geometric progression has first term a and com-
mon ratio r.
a. What is the formula for the nth term of this progression?
b. What is the formula for the sum of the first n terms of
this progression?
In Exercises 1–4, find the nth term of the arithmetic pro-
gression that has the given values of a, d, and n.
1. a 6, d 3, n 9 2. a 5, d 3, n 7
3. 4. a 1.2, d 0.4, n 98
5. Find the first five terms of the arithmetic progression
whose fourth and eleventh terms are 30 and 107, respec-
tively.
6. Find the first five terms of the arithmetic progression
whose seventh and twenty-third terms are 5 and 29,
respectively.
7. Find the seventh term of the arithmetic progression x, x y,
x 2y, . . . .
8. Find the eleventh term of the arithmetic progression a b,
2a, 3a b, . . . .
9. Find the sum of the first 15 terms of the arithmetic pro-
gression 4, 11, 18, . . . .
10. Find the sum of the first 20 terms of the arithmetic pro-
gression 5, 1, 7, . . . .
a 15, d
3
2
, n 8
11. Find the sum of the odd integers between 14 and 58.
12. Find the sum of the even integers between 21 and 99.
13. Find f (1) f(2) f (3) f(20), given that f(x)
3x 4.
14. Find t(1) t(2) t(3) t(50), given that t(x)
12 4x.
15. Show that Equation (19) can be written as
where a
n
represents the last term of an arithmetic progres-
sion. Use this formula to find:
a. The sum of the first 11 terms of the arithmetic pro-
gression whose first and eleventh terms are 3 and 47,
respectively.
b. The sum of the first 20 terms of the arithmetic progres-
sion whose first and twentieth terms are 5 and 33,
respectively.
16. S
ALES
G
ROWTH
Moderne Furniture Company had sales of
$1,500,000 during its first year of operation. If the sales
S
n
n
2
1a a
n
2
87533_05_ch5_p257-312 1/30/08 9:55 AM Page 306
increased by $160,000/year thereafter, find Moderne’s
sales in the fifth year and its total sales over the first 5 yr
of operation.
17. E
XERCISE
P
ROGRAM
As part of her fitness program, Karen
has taken up jogging. If she jogs 1 mi the first day and
increases her daily run by mi every week, when will she
reach her goal of 10 mi/day?
18. C
OST OF
D
RILLING
A 100-ft oil well is to be drilled. The
cost of drilling the first foot is $10.00, and the cost of
drilling each additional foot is $4.50 more than that of the
preceding foot. Find the cost of drilling the entire 100 ft.
19. C
ONSUMER
D
ECISIONS
Kunwoo wishes to go from the air-
port to his hotel, which is 25 mi away. The taxi rate is
$2.00 for the first mile and $1.20 for each additional mile.
The airport limousine also goes to his hotel and charges a
flat rate of $15.00. How much money will the tourist save
by taking the airport limousine?
20. S
ALARY
C
OMPARISONS
Markeeta, a recent college graduate,
received two job offers. Company A offered her an initial
salary of $48,800 with guaranteed annual increases of
$2000/year for the first 5 yr. Company B offered an initial
salary of $50,400 with guaranteed annual increases of
$1500/year for the first 5 yr.
a. Which company is offering a higher salary for the fifth
year of employment?
b. Which company is offering more money for the first
5 yr of employment?
21. S
UM
-
OF
-
THE
-Y
EARS
’-D
IGITS
M
ETHOD OF
D
EPRECIATION
One
of the methods that the Internal Revenue Service allows for
computing depreciation of certain business property is the
sum-of-the-years’-digits method. If a property valued at C
dollars has an estimated useful life of N years and a salvage
value of S dollars, then the amount of depreciation D
n
allowed during the nth year is given by
(0 n N)
where S
N
is the sum of the first N positive integers. Thus,
a. Verify that the sum of the arithmetic progression S
N
1 2 N is given by
b. If office furniture worth $6000 is to be depreciated by
this method over N 10 years and the salvage value of
the furniture is $500, find the depreciation for the third
year by computing D
3
.
22. S
UM
-
OF
-
THE
-Y
EARS
’-D
IGITS
M
ETHOD OF
D
EPRECIATION
Refer to Example 2, Section 1.3. The amount of deprecia-
tion allowed for a printing machine, which has an esti-
N1N 12
2
S
N
1 2 N
N1N 12
2
D
n
1C S2
N 1n 1 2
S
N
1
4
5.4 ARITHMETIC AND GEOMETRIC PROGRESSIONS 307
mated useful life of 5 yr and an initial value of $100,000
(with no salvage value), was $20,000/year using the
straight-line method of depreciation. Determine the
amount of depreciation that would be allowed for the first
year if the printing machine were depreciated using the
sum-of-the-years’-digits method described in Exercise 21.
Which method would result in a larger depreciation of the
asset in its first year of use?
In Exercises 23–28, determine which of the sequences
are geometric progressions. For each geometric progres-
sion, find the seventh term and the sum of the first seven
terms.
23. 4, 8, 16, 32, . . . 24.
25.
26. 0.004, 0.04, 0.4, 4, . . .
27. 243, 81, 27, 9, . . . 28. 1, 1, 3, 5, . . .
29. Find the twentieth term and sum of the first 20 terms of the
geometric progression 3, 3, 3, 3, . . . .
30. Find the twenty-third term in a geometric progression hav-
ing the first term a 0.1 and ratio r 2.
31. P
OPULATION
G
ROWTH
It has been projected that the popu-
lation of a certain city in the southwest will increase by 8%
during each of the next 5 yr. If the current population is
200,000, what is the expected population after 5 yr?
32. S
ALES
G
ROWTH
Metro Cable TV had sales of $2,500,000
in its first year of operation. If thereafter the sales increased
by 12% of the previous year, find the sales of the company
in the fifth year and the total sales over the first 5 yr of oper-
ation.
33. COLA
S
Suppose the cost-of-living index had increased by
3% during each of the past 6 yr and that a member of the
EUW Union had been guaranteed an annual increase equal
to 2% above the increase in the cost-of-living index over
that period. What would be the present salary of a union
member whose salary 6 yr ago was $42,000?
34. S
AVINGS
P
LANS
The parents of a 9-yr-old boy have agreed
to deposit $10 in their son’s bank account on his 10th birth-
day and to double the size of their deposit every year there-
after until his 18th birthday.
a. How much will they have to deposit on his 18th birth-
day?
b. How much will they have deposited by his 18th birth-
day?
35. S
ALARY
C
OMPARISONS
A Stenton Printing Co. employee
whose current annual salary is $48,000 has the option of
taking an annual raise of 8%/year for the next 4 yr or a
fixed annual raise of $4000/year. Which option would be
more profitable to him considering his total earnings over
the 4-yr period?
1
2
,
3
8
,
1
4
,
9
64
, . . .
1,
1
2
,
1
4
,
1
8
, . . .
87533_05_ch5_p257-312 1/30/08 9:55 AM Page 307
36. B
ACTERIA
G
ROWTH
A culture of a certain bacteria is known
to double in number every 3 hr. If the culture has an initial
count of 20, what will be the population of the culture at
the end of 24 hr?
37. T
RUST
F
UNDS
Sarah is the recipient of a trust fund that she
will receive over a period of 6 yr. Under the terms of the
trust, she is to receive $10,000 the first year and each suc-
ceeding annual payment is to be increased by 15%.
a. How much will she receive during the sixth year?
b. What is the total amount of the six payments she will
receive?
In Exercises 38–40, find the book value of office equip-
ment purchased at a cost C at the end of the nth year if
it is to be depreciated by the double declining-balance
method over 10 yr.
38. C $20,000, n 4 39. C $150,000, n 8
40. C $80,000, n 7
41. D
OUBLE
D
ECLINING
-B
ALANCE
M
ETHOD OF
D
EPRECIATION
Restaurant equipment purchased at a cost of $150,000 is to
be depreciated by the double declining-balance method
over 10 yr. What is the book value of the equipment at the
end of 6 yr? By what amount has the equipment been
depreciated at the end of the sixth year?
308
5 MATHEMATICS OF FINANCE
42. D
OUBLE
D
ECLINING
-B
ALANCE
M
ETHOD OF
D
EPRECIATION
Refer to Exercise 22. Recall that a printing machine with
an estimated useful life of 5 yr and an initial value of
$100,000 (and no salvage value) was to be depreciated. At
the end of the first year, the amount of depreciation
allowed was $20,000 using the straight-line method and
$33,333 using the sum-of-the-years’-digits method. Deter-
mine the amount of depreciation that would be allowed for
the first year if the printing machine were depreciated by
the double declining-balance method. Which of these three
methods would result in the largest depreciation of the
printing machine at the end of its first year of use?
In Exercises 43 and 44, determine whether the statement
is true or false. If it is true, explain why it is true. If it is
false, give an example to show why it is false.
43. If a
1
, a
2
, a
3
, . . . , a
n
and b
1
, b
2
, b
3
, . . . , b
n
are arithmetic
progressions, then a
1
b
1
, a
2
b
2
, a
3
b
3
, . . . , a
n
b
n
is also an arithmetic progression.
44. If a
1
, a
2
, a
3
, . . . , a
n
and b
1
, b
2
, b
3
, . . . , b
n
are geometric
progressions, then a
1
b
1
, a
2
b
2
, a
3
b
3
, . . . , a
n
b
n
is also a geo-
metric progression.
1. Use Equation (23) with a 24 and , obtaining
2. Use Equation (24) with C 75,000, N 5, and n 3,
giving the book value of the office equipment at the end of
3 yr as
or $16,200.
V132 75,000 a1
2
5
b
3
16,200
2411
1
32
2
3
2
33
2
S
5
2431 1
1
2
2
5
4
1 1
1
2
2
r
1
2
3. We have
S R R(1 i) R(1 i)
2
R(1 i)
n1
The sum on the right is easily seen to be the sum of the first
n terms of a geometric progression with first term R and
common ratio (1 i), so by virtue of Formula (23) we
obtain
S
R31 11 i2
n
4
1 11 i 2
R c
11 i 2
n
1
i
d
5.4 Solutions to Self-Check Exercises
Summary of Principal Formulas and Terms
CHAPTER 5
FORMULAS
1. Simple interest (accumulated amount)
2. Compound interest
a. Accumulated amount
b. Present value
P A11 i2
n
A P11 i2
n
A P11 rt2
87533_05_ch5_p257-312 1/30/08 9:55 AM Page 308
CONCEPT REVIEW QUESTIONS 309
Fill in the blanks.
1. a. Simple interest is computed on the
_____
principal
only. The formula for the accumulated amount using
simple interest is A
_____
.
b. In calculations using compound interest, earned interest
is periodically added to the principal and thereafter
itself earns
_____
. The formula for the accumulated
amount using compound interest is A
_____
. Solving
this equation for P gives the present value formula using
compound interest as P
_____
.
2. The effective rate of interest is the
_____
interest rate that
would produce the same accumulated amount in
_____
year
as the
_____
rate compounded
_____
times a year. The for-
mula for calculating the effective rate is r
eff
_____
.
3. A sequence of payments made at regular time intervals is
called a/an
_____
; if the payments are made at the end of
each payment period, then it is called a/an
_____ _____
; if
the payment period coincides with the interest conversion
period, then it is called a/an
_____ _____
.
4. The formula for the future value of an annuity is S
_____
;
The formula for the present value of an annuity is P
_____
.
5. The periodic payment R on a loan of P dollars to be amor-
tized over n periods with interest charged at the rate of i per
period is R
_____
.
6. A sinking fund is an account that is set up for a specific pur-
pose at some
_____
date. The periodic payment R required
to accumulate a sum of S dollars over n periods with inter-
est charged at the rate of i per period is R
_____
.
7. An arithmetic progression is a sequence of numbers in
which each term after the first is obtained by adding a/an
_____ _____
to the preceding term. The nth term of an
arithmetic progression is a
n
_____
. The sum of the first
n terms of an arithmetic progression is S
n
_____
.
8. A geometric progression is a sequence of numbers in which
each term after the first is obtained by multiplying the pre-
ceding term by a/an
_____ _____
. The nth term of a geo-
metric progression is a
n
_____
. If r 1, the sum of the
first n terms of a geometric progression is S
n
_____
.
c. Interest rate per conversion period
d. Number of conversion periods
3. Continuous compound interest
a. Accumulated amount
b. Present value
4. Effective rate of interest
5. Annuities
a. Future value
b. Present value
6. Amortization payment
7. Amount amortized
8. Sinking fund payment
R
iS
11 i 2
n
1
P R c
1 11 i 2
n
i
d
R
Pi
1 11 i 2
n
P R c
1 11 i 2
n
i
d
S R c
11 i 2
n
1
i
d
r
eff
a1
r
m
b
m
1
P Ae
rt
A Pe
rt
n mt
i
r
m
TERMS
simple interest (258) effective rate (263) ordinary annuity (276)
accumulated amount (258) present value (265) simple annuity (276)
compound interest (259) future value (265) future value of an annuity (278)
nominal rate (stated rate) (259) annuity (276) present value of an annuity (279)
conversion period (259) annuity certain (276) sinking fund (291)
Concept Review Questions
CHAPTER 5
87533_05_ch5_p257-312 1/30/08 9:55 AM Page 309
1. Find the accumulated amount after 4 yr if $5000 is invested
at 10%/year compounded (a) annually, (b) semiannually,
(c) quarterly, and (d) monthly.
2. Find the accumulated amount after 8 yr if $12,000 is in-
vested at 6.5%/year compounded (a) annually, (b) semi-
annually, (c) quarterly, and (d) monthly.
3. Find the effective rate of interest corresponding to a nomi-
nal rate of 12%/year compounded (a) annually, (b) semi-
annually, (c) quarterly, and (d) monthly.
4. Find the effective rate of interest corresponding to a nomi-
nal rate of 11.5%/year compounded (a) annually, (b) semi-
annually, (c) quarterly, and (d) monthly.
5. Find the present value of $41,413 due in 5 yr at an interest
rate of 6.5%/year compounded quarterly.
6. Find the present value of $64,540 due in 6 yr at an interest
rate of 8%/year compounded monthly.
7. Find the amount (future value) of an ordinary annuity of
$150/quarter for 7 yr at 8%/year compounded quarterly.
8. Find the future value of an ordinary annuity of $120/month
for 10 yr at 9%/year compounded monthly.
9. Find the present value of an ordinary annuity of 36 pay-
ments of $250 each made monthly and earning interest at
9%/year compounded monthly.
10. Find the present value of an ordinary annuity of 60 pay-
ments of $5000 each made quarterly and earning interest at
8%/year compounded quarterly.
11. Find the payment R needed to amortize a loan of $22,000
at 8.5%/year compounded monthly with 36 monthly in-
stallments over a period of 3 yr.
12. Find the payment R needed to amortize a loan of $10,000
at 9.2%/year compounded monthly with 36 monthly in-
stallments over a period of 3 yr.
13. Find the payment R needed to accumulate $18,000 with
48 monthly installments over a period of 4 yr at an interest
rate of 6%/year compounded monthly.
14. Find the payment R needed to accumulate $15,000 with
60 monthly installments over a period of 5 yr at an interest
rate of 7.2%/year compounded monthly.
15. Find the effective rate of interest corresponding to a nomi-
nal rate of 7.2%/year compounded monthly.
16. Find the effective rate of interest corresponding to a nomi-
nal rate of 9.6%/year compounded monthly.
17. Find the present value of $119,346 due in 4 yr at an inter-
est rate of 10%/year compounded continuously.
18. C
OMPANY
S
ALES
JCN Media had sales of $1,750,000 in the
first year of operation. If the sales increased by 14%/year
thereafter, find the company’s sales in the fourth year and
the total sales over the first 4 yr of operation.
19. CD
S
The manager of a money market fund has invested $4.2
million in certificates of deposit that pay interest at the rate of
5.4%/year compounded quarterly over a period of 5 yr. How
much will the investment be worth at the end of 5 yr?
20. S
AVINGS
A
CCOUNTS
Emily deposited $2000 into a bank
account 5 yr ago. The bank paid interest at the rate of
8%/year compounded weekly. What is Emily’s account
worth today?
21. S
AVINGS
A
CCOUNTS
Kim invested a sum of money 4 yr ago
in a savings account that has since paid interest at the rate
of 6.5%/year compounded monthly. Her investment is now
worth $19,440.31. How much did she originally invest?
22. S
AVINGS
A
CCOUNTS
Andrew withdrew $5986.09 from a
savings account, which he closed this morning. The
account had earned interest at the rate of 6%/year com-
pounded continuously during the 3-yr period that the
money was on deposit. How much did Andrew originally
deposit into the account?
23. M
UTUAL
F
UNDS
Juan invested $24,000 in a mutual fund
5 yr ago. Today his investment is worth $34,616. Find the
effective annual rate of return on his investment over the
5-yr period.
24. C
OLLEGE
S
AVINGS
P
ROGRAM
The Blakes have decided to
start a monthly savings program in order to provide for
their son’s college education. How much should they
deposit at the end of each month in a savings account earn-
ing interest at the rate of 8%/year compounded monthly so
that, at the end of the tenth year, the accumulated amount
will be $40,000?
25. R
ETIREMENT
A
CCOUNTS
Mai Lee has contributed $200 at
the end of each month into her company’s employee retire-
ment account for the past 10 yr. Her employer has matched
her contribution each month. If the account has earned
interest at the rate of 8%/year compounded monthly over
the 10-yr period, determine how much Mai Lee now has in
her retirement account.
26. A
UTOMOBILE
L
EASING
Maria has leased an auto for 4 yr at
$300/month. If money is worth 5%/year compounded
monthly, what is the equivalent cash payment (present
value) of this annuity? (Assume that the payments are
made at the end of each month.)
27. I
NSTALLMENT
F
INANCING
Peggy made a down payment of
$400 toward the purchase of new furniture. To pay the bal-
ance of the purchase price, she has secured a loan from her
bank at 12%/year compounded monthly. Under the terms
of her finance agreement, she is required to make payments
of $75.32 at the end of each month for 24 mo. What was
the purchase price of the furniture?
310
5 MATHEMATICS OF FINANCE
Review Exercises
CHAPTER 5
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BEFORE MOVING ON . . . 311
28. H
OME
F
INANCING
The Turners have purchased a house for
$150,000. They made an initial down payment of $30,000
and secured a mortgage with interest charged at the rate of
9%/year on the unpaid balance. (Interest computations are
made at the end of each month.) Assume the loan is amor-
tized over 30 yr.
a. What monthly payment will the Turners be required to
make?
b. What will be their total interest payment?
c. What will be their equity (disregard depreciation) after
10 yr?
29. H
OME
F
INANCING
Refer to Exercise 28. If the loan is amor-
tized over 15 yr:
a. What monthly payment will the Turners be required to
make?
b. What will be their total interest payment?
c. What will be their equity (disregard depreciation) after
10 yr?
30. S
INKING
F
UNDS
The management of a corporation antici-
pates a capital expenditure of $500,000 in 5 yr for the pur-
pose of purchasing replacement machinery. To finance this
purchase, a sinking fund that earns interest at the rate of
10%/year compounded quarterly will be set up. Determine
the amount of each (equal) quarterly installment that
should be deposited in the fund. (Assume that the pay-
ments are made at the end of each quarter.)
31. S
INKING
F
UNDS
The management of a condominium asso-
ciation anticipates a capital expenditure of $120,000 in 2 yr
for the purpose of painting the exterior of the condo-
minium. To pay for this maintenance, a sinking fund will
be set up that will earn interest at the rate of 5.8%/year
compounded monthly. Determine the amount of each
(equal) monthly installment the association will be re-
quired to deposit into the fund at the end of each month for
the next 2 yr.
32. C
REDIT
C
ARD
P
AYMENTS
The outstanding balance on Bill’s
credit-card account is $3200. The bank issuing the credit
card is charging 18.6%/year compounded monthly. If Bill
decides to pay off this balance in equal monthly install-
ments at the end of each month for the next 18 mo, how
much will be his monthly payment? What is the effective
rate of interest the bank is charging Bill?
33. F
INANCIAL
P
LANNING
Matt’s parents have agreed to con-
tribute $250/month toward the rent for his apartment in his
junior year in college. The plan is for Matt’s parents to de-
posit a lump sum in Matt’s bank account on August 1 and
then have Matt withdraw $250 on the first of each month
starting on September 1 and ending on May 1 the follow-
ing year. If the bank pays interest on the balance at the rate
of 5%/year compounded monthly, how much should
Matt’s parents deposit into his account?
Before Moving On . . .
CHAPTER 5
1. Find the accumulated amount at the end of 3 yr if $2000
is deposited in an account paying interest at the rate of
8%/year compounded monthly.
2. Find the effective rate of interest corresponding to a nomi-
nal rate of 6%/year compounded daily.
3. Find the future value of an ordinary annuity of $800/week
for 10 yr at 6%/year compounded weekly.
4. Find the monthly payment required to amortize a loan of
$100,000 over 10 yr with interest earned at the rate of
8%/year compounded monthly.
5. Find the weekly payment required to accumulate a sum of
$15,000 over 6 yr with interest earned at the rate of
10%/year compounded weekly.
6. a. Find the sum of the first ten terms of the arithmetic pro-
gression 3, 7, 11, 15, 19, . . . .
b. Find the sum of the first eight terms of the geometric
progression
1
2
, 1, 2, 4, 8, . . . .
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