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Chapter 1 FIN3414

# Chapter 1 FIN3414 - Intermediate Financial Management 1 2 3...

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Intermediate Financial Management 1 Chapter One 1 2 I: BASIC MATHEMATICAL TOOLS 3 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 = 7 64. In general, when we say Y n we mean multiply Y by itself n number of times. 8 9 Because of the extended use of exponents, we will briefly review the rules for dealing with 10 powers. 11 12 Rule 1: Y 0 = 1 Any number to the zero power is equal to one. 13 14 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 16 the exponents. This rule is also known as the product rule . 17 18 Rule 3: Y m / Y n = Y m-n The quotient of a common number or variable with different 19 exponents is just that number with a power equal to the 20 difference of the exponents. This rule is also called the quotient 21 rule . 22 23 Rule 4: ( Y m ) n = Y mn A variable or number with an exponent that is raised to another 24 power is equal to that number with a power equal to the product 25 of the exponents. This rule is also called the power rule . 26 27 Rule 5: Y 1/ n = n Y A variable or number that has an exponent of the form 1/ n is just 28 the n th root of that number. For instance, 36 1/2 is equivalent to the 29 square root of 36 or 2 36 = 36 =6. 30 31 Rule 6: Y -n = 1/ Y n A number or variable raised to a negative power is just the 32 reciprocal of that number raised to the positive power. 33 34 This last rule will come in very handy when working problems in the text. Few, if any, 35 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. 37 38 39

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Intermediate Financial Management 2 II.INTRODUCTION TO GEOMETRIC SERIES 40 41 A mathematical series is the sum of a sequence of real numbers. A series can be finite, 42 with limited number of terms, or infinite, with unlimited number of terms. For a finite series, let 43 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 44 expressed as 45 46 S n = a 1 + a 2 + …… + a n -1 + a n 47 = n i i a 1 48 49 Consider the following example: obtain the sum of 50 51 S = 1+ 2 + 2 2 + 2 3 + 2 4 52 53 By DIRECT METHOD, we have 54 55 S = 1+ 2 + 4 + 8 + 16 = 31. 56 57 However, consider the following: 58 59 S = 1+ 2 + 2 2 + 2 3 + 2 4. (I) 60 61 Multiplying both sides by 2, we get 62 63 2 S = 2+ 2 2 + 2 3 + 2 4 + 2 5 (II) 64 65 Now we subtract equation (I) from equation (II), we get 66 67 S = 2 5 1 = 31 68 69 What is unique about the above series? 70 71 For uniqueness, consider the following: 72 73 You pick any two consecutive terms, let us say 3 rd and 4 th , the third term is 2 2 and the fourth term 74 is 2 3 . Define the ratio, 75 76 erm precedingT Term succeeding R = 2 2 2 2 3 77 78 You can see that any succeeding term over preceding term remains constant and is equal to 2.
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Chapter 1 FIN3414 - Intermediate Financial Management 1 2 3...

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