quiz3sol - n=k 1 ∏ =-1 2 1 1 k i i = 1 1 k ∏ =-1 2 1 1...

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COT 3100 Quiz #3 Solutions Date: 11/21/00 1) Using induction, show that the following formula is true for all integers n 2. = - n i i 2 ) 1 1 ( = n 1 Use induction on n. Base case: n=2. LHS = 1 – ½ = ½, RHS = ½. Thus we have LHS = RHS. Inductive hypothesis: Assume for an arbitrary value of n=k that = - k i i 2 ) 1 1 ( = k 1 Inductive Step: Under this assumption we must show that the formula holds for
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Unformatted text preview: n=k+1: ∏ + =-1 2 ) 1 1 ( k i i = 1 1 + k ∏ + =-1 2 ) 1 1 ( k i i = [ ∏ =-k i i 2 ) 1 1 ( ] ( 1 – 1/(k+1)) = (1/k)*((k+1) – 1)/(k+1) = (1/k)*(k/(k+1)) = 1/(k+1), completing the inductive step. Thus, we can conclude that ∏ =-n i i 2 ) 1 1 ( = n 1 for all positive integers n ≥ 2....
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