Intermediate Financial Management
1
Chapter One
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2
I: BASIC MATHEMATICAL TOOLS
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4
As the reader will see, the study of the time value of money involves substantial use of
5
variables and numbers that are raised to a power. The power to which a variable is to be raised is
6
called an exponent. For instance, the expression 4
3
means four to the third power or 4 x 4 x 4 =
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64. In general, when we say
Y
n
we mean multiply Y by itself
n
number of times.
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Because of the extended use of exponents, we will briefly review the rules for dealing with
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powers.
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Rule 1:
Y
0
= 1
Any number to the zero power is equal to one.
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Rule 2:
Y
m
×
Y
n
=
Y
m+n
The product of a common number or variable with different
15
exponents is just that number with a power equal to the sum of
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the exponents.
This rule is also known as the
product rule
.
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Rule 3:
Y
m
/
Y
n
=
Y
mn
The quotient of a common number or variable with different
19
exponents is just that number with a power equal to the
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difference of the exponents.
This rule is also called the
quotient
21
rule
.
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Rule 4:
(
Y
m
)
n
= Y
mn
A variable or number with an exponent that is raised to another
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power is equal to that number with a power equal to the product
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of the exponents.
This rule is also called the
power rule
.
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Rule 5:
Y
1/
n
=
n
Y
A variable or number that has an exponent of the form 1/
n
is just
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the
n
th
root of that number. For instance, 36
1/2
is equivalent to the
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square root of 36 or
2
36
=
36
=6.
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Rule 6:
Y
n
= 1/
Y
n
A number or variable raised to a negative power is just the
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reciprocal of that number raised to the positive power.
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This last rule will come in very handy when working problems in the text. Few, if any,
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calculators will take the negative root of a number. However, by knowing that 5
3
is the same as
36
1/5
3
we can easily use a calculator to find that 5
3
= 125 and 1/125 equals 0.008.
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Intermediate Financial Management
2
II.INTRODUCTION TO GEOMETRIC SERIES
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A mathematical series is the sum of a sequence of real numbers.
A series can be finite,
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with limited number of terms, or infinite, with unlimited number of terms.
For a finite series, let
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n
be the number of terms in a series and
a
i
be the i
th
term of the series, then a finite series can be
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expressed as
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S
n
=
a
1
+
a
2
+ …… +
a
n
1
+
a
n
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=
n
i
i
a
1
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Consider the following example:
obtain the sum of
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S
= 1+ 2 + 2
2
+ 2
3
+ 2
4
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By DIRECT METHOD, we have
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S
= 1+ 2 + 4 + 8 + 16 = 31.
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However, consider the following:
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S
= 1+ 2 + 2
2
+ 2
3
+ 2
4.
(I)
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Multiplying both sides by 2, we get
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2
S
= 2+ 2
2
+ 2
3
+ 2
4
+ 2
5
(II)
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Now we subtract equation (I) from equation (II), we get
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S
= 2
5
–
1 = 31
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What is unique about the above series?
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For uniqueness, consider the following:
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You pick any two consecutive terms, let us say 3
rd
and 4
th
, the third term is 2
2
and the fourth term
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is 2
3
.
Define the ratio,
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erm
precedingT
Term
succeeding
R
=
2
2
2
2
3
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You can see that any succeeding term over preceding term remains constant and is equal to 2.
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 Fall '10
 CHAIYUTHPADUNGSAKSAWASDI
 Finance, Time Value Of Money, Logarithm, Intermediate Financial Management

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