Discrete Mathematics with Graph Theory (3rd Edition) 162

# Discrete Mathematics with Graph Theory (3rd Edition) 162 -...

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160 Solutions to Review Exercises which is formula (*) with n = k + 1. By the Principle of Mathematical Induction, the formula given in (*) holds for all n 2:: 2. (b) The values of the given product, for n = 2,3,4,5 are, respectively, ~, ~ = t, i, ~ = 1 6 0' so we guess that 1 1 1 1 n+l (1 - -)(1 - -)(1 - -) ... (1 - -) = - 4 9 16 n 2 2n for n 2:: 2. For n = 2, the left side is 1 - ~ = ~, which is the right side, so the formula is correct. Now assume that k 2:: 2 and 1 1 1 1 k+l (1- 4)(1- 9)(1- 16)'" (1- k 2 ) = """2k' With n = k + I, the product is 1 1 1 1 (1 - 4)(1- 9)(1- 16)'" (1 - (k + 1)2) 1 1 1 1 1 = [(1- 4)(1- 9)(1- 16)'" (1- k2)](I- (k + 1)2) k+l 1 = 2k""(1 - (k + 1)2)' using the induction hypothesis, k + 1 1 = """2k - 2k(k + 1) (k+l)2-1 k2+2k k(k+2) k+2 2k(k + 1) 2k(k + 1) 2k(k + 1) 2(k + 1) , which is formula (**) with n = k + 1. By the Principle of Mathematical Induction, the formula given in (**) holds for all
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Unformatted text preview: n 2:: 2. 11. The sequence is 47,41, 35, 29, ... , which is arithmetic with a = 47 andd = -6. So S = ~[2a+ (n-l)d] = 50[94 + 99( -6)] = -25000. 12. This is the sum of an arithmetic sequence with a = 529, d = -4 and an = -459. Since an = a + (n -l)d, we obtain n = 498. The sum is n 498 S = 2[2a + (n -1)d] = 2[1058 - 4(497] = 249( -930) = -231,570. l3. For n = I, the formula gives ~(1 -(_!)O) = ~(1 -1) = 0 = al. For n = 2, the formula gives ~(1 -(_!)l) = ~(1 + !) = l = a2. Thus the result is true for n = 1 and n = 2. Now assume that k > 2 and that the formula is correct for all n < k. We wish to prove that the result is true when n = k. We have ak = !(ak-2 + ak-l) (given) = ! [~(1 -(_!)k-3) + ~(1 _ (_!)k-2)] (the induction hypothesis) _ ! [~ _ ~(_!)k-3 + ~ _ ~ (_!)k-2] -29 9 2 9 9 2 = H~) [2 -(_!)k-3(1 -!)] = H~) [2 + (_!)k-3(_!)] = ~(1-(_!)k-l)...
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## This note was uploaded on 11/08/2010 for the course MATH discrete m taught by Professor Any during the Summer '10 term at FSU.

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