AP 03 - Jan 23

AP 03 - Jan 23 - Math 2602 M1/M2 Spring 2007 Additional...

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Unformatted text preview: Math 2602 M1/M2 Spring 2007 Additional Problems 3 The Principle of Mathematical Induction 21 Thursday, Jan 23th 1. Use the Principle of Mathematical Induction to show that for positive integer n and k n+k+1 k n+k+1 k = = n+k n+k-1 n+2 n+1 n + + + + + . k k-1 2 1 0 n+k n+k-1 n+2 n+1 n + + + + + . n n n n n k k+1 + . 2 2 2. (a) For k Z+ verify that k2 = (b) Fix n Z+ . Since the result in part (a) is true for all k = 1, 2, . . . , n, summing the n equations 12 = 22 = . . . n2 = n n+1 + 2 2 n+1 n+2 + . 3 3 1 2 + 2 2 2 3 + 2 2 and using the previous exercise show that n k2 = k=1 n(n + 1)(n + 2) = 6 (c) Prove part (b) by induction. (d) For k Z+ verify that k3 = k k+1 k+2 +4 + . 3 3 3 n+1 n+2 n+3 +4 + . 4 4 4 Using the previous exercise show that n k3 = k=1 n2 (n + 1)2 = 4 (e) Show part (d) by induction (f) Find a, b, c, d Z+ so that for k Z+ k4 = a k k+1 k+2 k+3 +b +c +d . 4 4 4 4 (Answer: a = 1, b = 11, c = 11, d = 1.) For more info look for W orpitzky s identity. 1 T. Yolcu, School of Mathematics at Georgia Tech. Email: tyolcu@math.gatech.edu. 1 ...
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