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CoraGB f(n) = [1-(1/2^2)]*[1-(1/3^2)]*...*[1-(1/n^2)] where f: N − {1}→Q
I need to use some small values of n to come up with a conjecture of F(n) and the prove it using induction.
So far I have:
Once I calculated some values I conjectured that f(n)=(n+1)/2n
So my basis is n=2, ([1-(1/2^2)] = 3/4 = (2+1)/2*2
Hypothesis is n=k, then [1-(1/2^2)]*[1-(1/3^2)]*...*[1-(1/k^2)] = (k+1)/2k
Induction Step: n=k+1,
[1-(1/2^2)]*[1-(1/3^2)]*...*[1-(1/k^2)]*[1-(1/(k+1)^2)] = [(k+1)+1]/2(k+1)
So the right side becomes k+2/(2k+2)
And the left side becomes k+1/2k*[1-(1/(k+1)2)]
I tried simplifying the left side but I am definitely not getting anything similar to the right side. Any advice?