Hello,

I know this is a pretty basic question for you guys but I'm trying to learn induction and I need to see how this done please help with this problem...

20 + 21 + 22 +... + 2n = 2n+1 -1 is true whenever n is a positive integer.

Thanks for your help Kyle

Kyle,

Let p(n) be the statement that 20 + 21 + 22 +... + 2n = 2n+1 -1. You need to first check this out for n = 1, i.e. 20 + 21 = 3 = 21+1 -1. Next assume that you have checked it out for n = 1, ..., k for some k > = 1. In particular you now know that 20 + 21 + 22 + ...+ 2k = 2k+ 1 -1, i.e. p(k) holds - this is the induction hypothesis . What is necessary to complete the proof is to show that this knowledge implies p(k+1) holds.

Now 20 + 21 + 22 + ...+ 2k+1 = 20 + 21 + 22 + ...+ 2k + 2k+1 = 2k+1 -1 + 2k+1 (using the induction hypothesis) = 2x2k+1 -1 = 2k+2 -1 as required. Thus p(k) holds implies that p(k+1) holds and since p(1) is true, p(n) is true for all n >= 1.

Penny

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