ISI2019-MMA-26

Question

If $t = \begin{pmatrix} 200 \\ 100 \end{pmatrix}/4^{100} $, then

• A. $t < \frac{1}{3}$
• B. $\frac{1}{3} < t < \frac{1}{2}$
• C. $\frac{1}{2} < t < \frac{2}{3}$
• D. $\frac{2}{3} < t < 1$

Answer 1

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.

Comments

Even $f(3)=\frac{5}{16}<\frac{1}{3}$ will suffice.

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Focus on the main idea i.e. it is a decreasing sequence.