Induction ProofDate: 10/15/97 at 01:56:48 From: Masika bryce Subject: Induction proof. This induction problem asks us to find the formula by examining the values of the expression 1/1.2 + 1/2.3 + .... + 1/n(n+1), and to prove the result. I have set up a formula: => 2^n-1/n(n+1) and have tried to prove it is true. I could not continue because I did not really know how to substitute for P(n+1). Please help. Date: 10/15/97 at 08:37:14 From: Doctor Jerry Subject: Re: Induction proof. Hi Masika, I think your formula is not correct. Here's a way of finding a correct formula in this case. First, notice that 1/(n(n+1)) = 1/n - 1/(n+1). Then, 1/(1.2)+1/(2.3)+...+1/(n(n+1)) = 1-1/2+1/2-1/3+1/3-1/4+...+1/n-1/(n+1) You can see that there are many cancellations, leaving 1/(1.2)+1/(2.3)+...+1/(n(n+1)) = 1-1/(n+1) To prove this by induction: 1. Check to see if it's okay for n = 1. 1/(1.2) = 1-1/2. Okay. 2. Assume true for n=k, so that 1/(1.2)+...+1/(k(k+1))=1-1/(k+1), and try to prove that it's true for n = k+1, that is, 1/(1.2)+1/(2.3)+...+1/((k+1)(k+2)) = 1-1/(k+2). You can do this by just adding 1/((k+1)(k+2)) to both sides of the assumed case. -Doctor Jerry, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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