w8 - Math 55 Worksheet Adapted from worksheets by Rob...

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Math 55 Worksheet Adapted from worksheets by Rob Bayer, Summer 2009. Induction 1. Prove that the sum of the first n odd numbers is n 2 . 2. Using the product rule d dx ( f ( x ) g ( x )) = f 0 ( x ) g ( x ) + f ( x ) g 0 ( x ), and the fact that d dx x = 1, use induction to prove that d dx ( x n ) = nx n - 1 . 3. Let a n be the sequence defined by a 1 = 2, a n = 2 a n - 1 . (a) Show that 1 < a n < 2 for all n . (b) Show that a n +1 > a n for all n . NOTE: A sequence like this is called bounded and monotone and is guaranteed to converge. 4. The harmonic numbers are defined by H n = 1 + 1 2 + 1 3 + 1 4 + ··· + 1 n . (a) Show that H 2 n 1 + n 2 for all n . (b) Use your answer to (a) to show that X n =1 1 n increases without bound. 5. Prove that 1 + 2 3 + 3 3 + 4 3 + ··· + n 3 = ± n ( n + 1) 2 ² 2 . Strong Induction and Well-ordering 1. Prove that if n 18, then you can make n cents out of just 4- and 7-cent stamps 2. Here we’ll use well-ordering to show that x 2 + y 2 = 3
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