Re: - - Proof by mathematical induction

Tim Little <tim@xxxxxxxxxxxxxxxxxx> wrote:
On 2009-09-29, qsymmetry <qsymmetry@xxxxxxxxx> wrote:

For each natural n, let f(n) = (n^2 + n + 1)^2 + 1 and let

P_n = [ f(1) f(3) ... f(2n - 1)]/ [f(2) f(4) ... f(2n)]

I would like to show that P_1 + .. + P_n < 1/2 for each n.

I've shown that P_1 and P_1 + P_2 < 1/2; and I also noticed that

f(2n - 1)/f(2n) < 1

but I don't see how this observation helps in proving the above
inequality.

It doesn't. However, the expression for P_n in terms of n simplifies
surprisingly well,

HINT f(n) = g(n) g(n+1) so multiplicative telescopy ensues

and the inequality should be much easier to prove from the simplified form.

.

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