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Unformatted text preview: + (D. Now assume that k > 2 and the result is true for all n in the range 1 ~ n < k. We wish to prove the result for k. Note that the identity we wish to prove has two different types of ~ final terms depending on whether n is even or odd. When n is odd, the final term in the sum is (n~l) 2 while, if n is even, the final term is (%). We establish the odd and even cases separately. 2 Case 1: n is odd. ( n) (n 1) (n 2) (!!l) In this case, we wish to prove that an = 0 + 1 + 2 + . .. + n~l . (Note how identity (5) was employed at the very first step.) Now applying the induction hypothesis to n  1, the term in the first set of brackets here is anl and applying the induction hypothesis to n  2, the other term is a n 2. Case 2: n is even. ( n) (n 1) (n 2) (!!) This time we wish to prove an = 0 + 1 + 2 + . .. + ; ....
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 Summer '10
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 Graph Theory

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