Proving $\prod((k^2-1)/k^2)=(n+1)/(2n)$ by induction $$P_n = \prod^n_{k=2} \left(\frac{k^2 - 1}{k^2}\right)$$ Someone already helped me see that $$P_n = \frac{1}{2}.\frac{n + 1}{n} $$  Now I have to prove, by induction, that the formula for $P_n$ is correct. The basis step: $n = 2$ is true, $$P_2 = \frac{3}{4} = \frac{1}{2}.\frac{2+1}{2} $$  Inductive step: Assuming $P_k = \frac12.\frac{k+1}k$ is true for $k \ge2$, I need to prove $$P_{k+1} = \frac12.\frac{k+2}{k+1}$$  So I am stuck here, I have been playin around with $P_k$ and $P_{k+1}$ but I can't figure out how to connect the hypothesis with what I am trying to prove.

We do the same thing as in the solution of Brian M. Scott, but slightly backwards, We are interested in the question 
$$\frac{n+2}{2(n+1)}\overset{?}{=}\frac{n+1}{2n}\frac{(n+1)^2-1}{(n+1)^2}.$$
The difference-of-squares factorization $(n+1)^2-1=(n)(n+2)$ settles things. 
