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mt2 - d and let b 1,b n be a geometric progression with a...

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Homework 2, Math 245, Fall 2010 Due Wednesday, September 22 in class. The solution of each exercise should be at most one page long. If you can, try to write your solutions in LaTex. Each question is worth 2 points. 1. In a tournament with n 2 teams, every team plays against every other team exactly once. Prove that the number of games played is n ( n - 1) 2 . 2. Find and prove a simple formula for the following sum: 1 · x + 2 · x 2 + · · · + n · x n . 3. If x 0, show that (1 + x ) n 1 + nx + n ( n - 1) 2 x 2 for any n 2. 4. Let a 1 , . . . , a n , . . . be an arithmetic progression with a non-zero common difference
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Unformatted text preview: d , and let b 1 ,...,b n ,... be a geometric progression with a positive ratio r . Prove that there exist numbers x and y such that b n = xy a n for all n ≥ 1. 5. Assume that the numbers x 1 ,x 2 ,...,x n are all non-zero and form an arithmetic progres-sion. Show that 1 x 1 x 2 + ··· + 1 x n-1 x n = n-1 x 1 x n for any n ≥ 2. 6. Bonus Question If the equality above is true for all n ≥ 2, does it imply that the sequence ( x n ) n ≥ 1 is an arithmetic progression ?...
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