Unformatted text preview: _xk+l = _(x k . x) = (_xk) . X + xkx = kx kl . X + xk . 1 = kxk + xk = (k + l)xk dx dx dx dx as required. By the Principle of Mathematical Induction, we conclude that the formula is correct for all n 2: 1. r 1 X 2r + 1 1 _ x4 34. When n = 1, rr;=l (1 + x 2 ) = 1 + x 2 and 2 = 2 = 1 + x 2 and so the formula is Ix Ix correct. Now suppose that k 2: 1 and the result is true for n = k; that is, suppose k 2k+1 II( 1 2r)=Ix +x 1 2 x r=l We must prove that it is correct for n = k + 1; that is, we must prove that k+l Now II (1 + x2r) r=l k = (II (1 + x2r)) (1 + X 2k + 1 ) r=l 2 k + 1 _ (11_Xx2 ) (1 + x 2k + 1 ) (by the induction hypothesis) as required. By the Principal of Mathematical Induction, we conclude that the formula is correct for all n 2: 1....
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 Summer '10
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 Logic, Graph Theory, Product Rule, Mathematical Induction, induction hypothesis

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