Mathematical Induction Question

[XJellieBX]

Find a simplifying expression for the product
(1-1/2^{2})(1-1/3^{2})...(1-1/n^{2})
and verify its validity for all integers n \geq 2

I know how to do the second part of the question, but no idea how to approach the simplifying of the expression. Any tips?

 

[jacksonpeeble]

I'm afraid I can't help you with an exact answer, but perhaps you could apply sigma in this situation?

I'm not positive though, so please don't go all out with sigma, there's a high chance that I'm wrong. Hopefully one of the geniuses on this site will be along shortly to help you more!

 

[HallsofIvy]

I am puzzled as to how you can do the second part of the problem (verify that the given expression is equal to the simplified expression) without knowing what the simplified expression is!

If I were looking for a simplified expression, I think I would start by writing out a few terms: (1- 1/4)(1- 1/9)(1- 1/16)...= (3/4)(8/9)(15/16)... which makes it easy to see that each term is of the form (n²-1)/n². Furthermore n²- 1= (n-1)(n+1) so we should be able to cancel each numerator with part of the previous and next denominator.