FRACtorial !

Algebra Level 3

If, [ 2 × ( 1 2 ) ! ] 2 ( − 1 2 ) ! = S \cfrac{\left[2×\left(\dfrac{1}{2}\right)!\right]^2}{\left(\dfrac{-1}{2}\right)!}=\sqrt{S} Then find, ⌊ 10000 S ⌋ \lfloor10000S\rfloor


The answer is 31415.

Sid 2108 by Sid 2108 , 21, India

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1 solution

Sid 2108
Mar 9, 2015

Now ( 1 2 ) ! = π 2 \left(\dfrac{1}{2}\right)!=\dfrac{\sqrt{\pi}}{2}

And ( − 1 2 ) ! = π \left(\dfrac{-1}{2}\right)!=\sqrt{\pi}

Therefore. [ 2 × ( 1 2 ) ! ] 2 ( − 1 2 ) ! \cfrac{\left[2×\left(\dfrac{1}{2}\right)!\right]^2}{\left(\dfrac{-1}{2}\right)!}

Becomes [ 2 × ( π 2 ) ] 2 π = π \cfrac{\left[2×\left(\dfrac{\sqrt{\pi}}{2}\right)\right]^2}{\sqrt{\pi}}=\sqrt{\pi}

Thus S = π S=\pi

Therefore ⌊ 10000 S ⌋ = 31415 \boxed{\lfloor10000S\rfloor=31415}

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