Find the value of x

Question

$$\left(1-\frac{1}{2^2}\right)\left(1-\frac{1}{3^2}\right)\left(1-\frac{1}{4^2}\right)\ldots\left(1-\frac{1}{2011^2}\right)=\frac{x}{2\cdot 2012}$$

Help me solve this, obviously there is a shortcut to solve it, but I have tried a lot and I've come up with something confusing!

Did you try rewriting the factors along the lines of $1-1/n^2=(n^2-1)/n^2=(n-1)(n+1)/n^2$? It looks to me like you could get some cancellation going … — Harald Hanche-Olsen, Oct 26 '14 at 19:41
yea, I wrote this way, but actually it's not way of solving this — Erik Hambardzumyan, Oct 26 '14 at 19:46
Something about the writing of this feels like a competition math problem. — Asimov, Oct 26 '14 at 20:12
When you cancel it, the answer should be $\frac{x+2}{2x+2}$ where x is the number of terms. So, in this case, x=2011, so the numerator is x+2=2013. — Pakquebchsoflwty, Oct 26 '14 at 20:13
@Asimov I think this is (at least) from an Estonian math competition from 2010, with different numbers, maybe. Can't find a link, but I was in that competition. — gtrrebel, Oct 27 '14 at 19:38
@gtrrebel I would have guessed from 2011, or 2012. The numbers they use are never random. — Asimov, Oct 27 '14 at 22:12

Mufasa · Accepted Answer · 2014-10-26T21:51:19.167

1

If you use the hint (given by @Harald Hanche-Olsen) and write out some terms you will see a pattern emerging:$$\require{cancel}\frac{1.\cancel{3}}{2.\cancel{2}}\frac{\cancel{2}.\cancel{4}}{\cancel{3}.\cancel{3}}\frac{\cancel{3}.5}{\cancel{4}.4}...$$Can you see how all the intermediate terms will cancel out?

edited Oct 26 '14 at 21:51

answered Oct 26 '14 at 20:05

Mufasa

5,434

BTW: I tried using the latext \cancel{3} notation to show the terms that would cancel but it doesn't seem to work anymore - is this a new bug? – Mufasa Oct 26 '14 at 20:06
1

No, you have to put \require{cancel} after the first $. – rae306 Oct 26 '14 at 20:09
Perhaps split it up as $\cfrac 12\cdot \cfrac 32\cdot \cfrac 23\cdot \cfrac 43 \cdot \cfrac 34\cdot \cfrac 45 \dots $ – Mark Bennet Oct 26 '14 at 20:10
@RainiervanEs - thanks for the latex clarification - I have used it to show how cancellation would cascade through the terms – Mufasa Oct 26 '14 at 21:52

Find the value of x

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