Re: Induction problems
From: José Carlos Santos (jcsantos_at_fc.up.pt)
Date: 10/30/04
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Date: Sat, 30 Oct 2004 20:57:04 +0100
Neil L. wrote:
> Here's the whole problem. Prove that 3^n > 20n, for each integer n >= 4.
>
> so, n=4, 3^4 = 81 and 20n = 80, so 3^n > 20n
>
> Now, assume induction hypothesis 3^k > 20k, for each integer k >= 4. We
> must prove 3^(k+1) > 20(k+1) We have:
>
> 3^(k+1) = 3(3^k)
> 3^(k+1) > 3(20k) (by the induction hypothesis)
> 3^(k+1) > 20k + 40k
> 3^(k+1) > 20k + 20 (since k >= 4 > 1/2)
> 3^(k+1) > 20(k+1)
>
> The 2nd last step is where my issue comes in. I DONT GET IT!!!
The second step? The one that says that "3^(k+1) > 3(20k)"? Well,
since 3^k > 20k (by the induction hypothesis), then 3*3^k > 3*(20k).
Best regards,
Jose Carlos Santos
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