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Unformatted text preview: (11)4 n + (11)4 n4 n +1 = ((11) n4 n )(11) + 4 n (114) . Since (11) n4 n is divisible by 7, and 4 n (114) = 7(4) n is divisible by 7, we have that (11) n +14 n +1 is divisible by 7. Therefore, P n implies P n +1 so the induction step holds, and the principle of mathematical induction implies that P n is true for all n . Polyas Paradox. The induction step does not hold for n = 1. If there are n + 1 = 2 horses, then the induction step asks us to partition { 1 , 2 } into the sets { 1 } and { 2 } which do not overlap. The rst base case for which the induction step works is n = 2, but then the statement of the theorem is obviously false: Two horses can have two dierent colors. 1...
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 Spring '09

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