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s8 - Math 55 Worksheet Adapted from worksheets by Rob Bayer...

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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 . Solution: Try this one on your own! 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 . Solution: We are inducting on n . So the base case is when n = 1. Since d dx ( x 1 ) = 1 x 0 the base case is true. So assume that d dx ( x k ) = kx k - 1 . We want to show that d dx ( x k +1 ) = ( k + 1) x k . Use the product rule on x · x k - 1 and now use the inductive hypothesis. You’ll learn much more by finishing this yourself, than by reading a solution. If you get stuck, you can email me! 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 . Solution: We’ll use induction. The base case is that 1 < a 1 < 2 this is true. Now assume that 1 < a k < 2. We want to show that 1 < a k +1 < 2. That is, we want to show that a < 2 a k < 2 . But now, think about 2 a k .... Since we’re assuming 1 < a k < 2 we know that 2 < 2 a k < 4. Thus 2 a k must be between 2 and 4. So it’s definitely between 1 and 2. (b) Show that a n +1 > a n for all n . Solution: Ok, so we want to show that 2 a k > a k . Since a k is positive, we can divide through by a k , and it’s equivalent to show that 2 > a k . But this is true by part a ) . I guess we didn’t need induction.
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