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ISI2019-MMA-26

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Consider the function$f(n) = \binom{2n}{n}/4^n $

So $f(n+1) = \binom{2n+2}{n+1}/4^{n+1}$

Dividing the two and  after solving, we get

$f(n+1)/f(n) = (n+1/2)/(n+1) <1 \forall n>1$

That means $f(1)>f(2)>f(3)...>f(100)$

$\implies f(100)< f(4) = 70/256 < 1/3$

Hence $\textit A$ is the correct answer.
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