Unformatted text preview: i=l k k k+1 k+1 = [(LXi) +Xk+1] + [(LYi) +Yk+1] = LXi + LYi, i=l i=l i=l i=l which is the formula with n = k + 1. Thus, the result holds for all n ;::: 1 by the Principle of Mathematical Induction. (b) For n = 1, 2:~=1 CXi = 2::=1 CXi = CX1 = C 2::=1 Xi and the formula is true. Now suppose that k ;::: 1 and the formula is true for n = k. Then k+1 k k L CXi = (L CXi) + CXk+1 = (c L Xi) + CXk+1, (by the induction hypothesis) i=l i=l i=l k k+1 =C[(LXi) +Xk+1] =CLXi, i=l i=l which is the formula with n = k + 1. Thus, the result holds for all n ;::: 1 by the Principle of Mathematical Induction. n 2 (c) For n = 2, L(Xi XiI) = L(Xi XiI) which is, by convention, the single term X2 X21 = i=2 i=2 X2 Xl = Xn Xl. Thus, the formula is true for n = 2. Now suppose that k ;::: 2 and the formula...
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 Summer '10
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 Graph Theory, Mathematical Induction, Inductive Reasoning, Natural number, Mathematical logic, Mathematical proof

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