Let n > 1 be a positive integer. Prove combinatorially that n = n(n+ 1)2-2. k...

  

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Let n > 1 be a positive integer. Prove combinatorially that n = n(n+ 1)2"-2. k k=1 [Hint 1: Suppose that out of n people, anyone is permitted to register for a race. Everyone who registers may run twice. Gold and Silver medals are to be awarded, and the same person can win both medals. Everyone else gets a participation trophy.] [Hint 2: n(n + 1) = n(n – 1) + 2n] Let n > 1 be a positive integer. Prove combinatorially that n = n(n+ 1)2"-2. k k=1 [Hint 1: Suppose that out of n people, anyone is permitted to register for a race. Everyone who registers may run twice. Gold and Silver medals are to be awarded, and the same person can win both medals. Everyone else gets a participation trophy.] [Hint 2: n(n + 1) = n(n – 1) + 2n]