M147A1

# M147A1 - MATH 147 Assignment #1 Due: Friday September 25 1)...

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MATH 147 Assignment #1 Due: Friday September 25 1) Prove that: a) 1 2 + 2 2 + ··· + n 2 = n ( n +1)(2 n +1) 6 for each n N . b) 2 n + 3 n is divisible by 5 for each odd n N (Hint: Odd n ’s are of the form 2 k - 1 for k N . 2) Let a 1 = 1 and for each n 1 let a n +1 = 3 + 2 a n (This is called a recursively deﬁned sequence.) Prove that for every n N , we have 0 a n a n +1 3 3) Tower of Hanoi: You are given three pegs. On one of the pegs is a tower made up of n rings placed on top of one another so that as you move down the tower each successive ring has a larger diameter than the previous ring. The object of this puzzle is to reconstruct the tower on one of the other pegs by moving one ring at a time, from one peg to another, in such a manner that you never have a ring above any smaller ring on any of the three pegs. Prove that for any n N , if you begin with n rings, then the puzzle can be completed in 2 n - 1 moves. Moreover, prove that for each

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## This note was uploaded on 10/21/2010 for the course MATH 147 taught by Professor Wolzcuk during the Fall '09 term at Waterloo.

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M147A1 - MATH 147 Assignment #1 Due: Friday September 25 1)...

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