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THE SECOND PRINCIPLE OF MATHEMATICAL INDUCTION The Second Principle of Mathematical Induction, also called strong induction, is nearly the same as the first principle. The only difference is in the inductive step. Theorem A.4 The Second Principle of Mathematical Induction If {P1 , P2, P3 , . . . } is a sequence of statements such that: P1 is true. Each statement Pn can be deduced from its predecessors {P1 , P2, P3 , . . . , Pn 1 }. Then all of the statements P1, P2 , P3, . . . are true. So to verify the inductive step with strong induction, we may assume that all n 1 statements P1 , P2 , P3, . . . , Pn 1 are true. EXAMPLE A.5 Strong Induction
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Prove that the Fibonacci numbers 0, 1, 1, 2, 3, 5, 8, 13, 21, . . . are asymptotically exponential. More precisely, we prove that Fn = O(2 n ), where the Fibonacci numbers Fn are defined as F0 = 0, F1 = 1, and Fn = Fn 1 + Fn 2. So our sequence of statements is P1 : P2 : F1 F2 21 22
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P3 : F3 23 etc. These first few are true because of the following relationships: P1 : F1 = 1 2 P2 : F2 = 2 4 P3 : F3 8 For the inductive step, we assume that n 1 statements P1 , P2, P3 , . . . , Pn 1 are true and compare them with the nth statement Pn: P1 : P2 : P3 : : Pn 2: Pn 1: F1 F2 F3 : Fn 2 Fn 1 2n 2 2n 1 21 22 23
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Pn : Fn 2n Comparing the nth statement with the two that precede it, we see that Fn = Fn 1 + Fn 2 2n = (2)(2n 1) = 2n 1 + 2n 1 > 2n 1 + 2n 2 So we can derive Pn from Pn 1 and Pn 2 like this: Fn = Fn 1 + Fn 2 2n 1 2n 2 < 2n 2n for all n). This proves that all the statements are true (i.e., Fn
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A series is a sequence of possibly infinitely many terms whose sum is to be determined. A geometric series is a series in which each term is the same multiple of its predecessor. For example, 20 + 60 + 180 + 540 + 1620 + 4860 + is a geometric series because each term is 3 times the size of its predecessor. The multiplier 3 is called the common ratio of the series. Theorem A.5 Sum of a Finite Geometric Series If r 1, then a 1 rn = --------------------1 r Here, a is the first term in the series, r is the common ratio, and n is the number of terms in the series. a + ar + ar + ar +
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+ ar
EXAMPLE A.6 Finite Geometric Series
For the sum 20 + 60 + 180 + 540 + 1620 + 4860, the three parameters are a = 20, r = 3, and n = 6. So the sum is a 1 r 20 1 3 20 1 729 20 729 -------------------- = ------------------------ = ---------------------------- = ---------------------- = 7280 1 r 1 3 2 2
Theorem A.6 Sum of an Infinite Geometric Series If 1 < r < 1, then a + ar + ar + ar + EXAMPLE A.7 Infinite Geometric Series
For the sum 0.42 + 0.0042 + 0.000042 + 0.00000042 + 0.0000000042 + a = 0.42 and r = 0.01. So the infinite sum is a0.42 ------------------ = ------------------ = 0.42 = 42 = 14 --------1 r 1 0.01 0.99 99 33 , the three parameters are