The product of $(1-1/(2^2))(1-1/(3^2))....(1-1/(9^2))(1-1/(10^2))=a/b$ then find a and b without full simplification?

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Answer 1 · 2018-07-17T16:31:49

The answer is $a=11$ and $b=20$

I think it's the same as

$Pi_(k=2)^10(1-1/k^2)$

But I did it this way

$(1-1/2^2)(1-1/3^2)(1-1/4^2)....(1-1/9^2)(1-1/10^2)$

$=3/4xx8/9xx15/16xx24/25xx35/36xx48/49xx63/64xx80/81xx99/100$

$=11/20$

The answer is $a=11$ and $b=20$

Answer 2 · 2018-07-17T19:45:32

$(1-1/2^2)(1-1/3^2)(1-1/4^2)....(1-1/9^2)(1-1/10^2)$

$=(1-1/2)(1+1/2)(1-1/3)(1+1/3)(1-1/4)(1+1/4)....(1-1/9)(1+1/9)(1-1/10)(1+1/10)$

$=1/2*11/10=11/20$

So $a=11andb=20$

The product of (1-1/(2^2))*(1-1/(3^2))....(1-1/(9^2))*(1-1/(10^2))=a/b(1-1/(2^2))*(1-1/(3^2))....(1-1/(9^2))*(1-1/(10^2))=a/b then find a and b without full simplification?