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Question #40635

Difficulty: 8/10 | 🟢 Free

The Question

Let $M_n$ be the greatest binomial coefficient in the expansion of $(1+x)^{2n}$ . Which of the following inequalities holds strictly true for all integers $n \ge 1$ ?

Options

A.
$\frac{16^n}{\pi(n+1)} < M_n^2 < \frac{16^n}{\pi(n+\frac{1}{2})}$
B.
$\frac{16^n}{\pi(n+\frac{1}{2})} < M_n^2 < \frac{16^n}{\pi n}$
C.
$\frac{16^n}{\pi n} < M_n^2 < \frac{16^n}{\pi(n-\frac{1}{2})}$
D.
$\frac{16^n}{\pi(n+\frac{1}{4})} < M_n^2 < \frac{16^n}{\pi(n-\frac{1}{4})}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Greatest Binomial Coefficient|Wallis' Formula|Inequalities in Definite Integrals

Why This Question Is Tricky

Common Mistakes

This problem tests the student's ability to bridge discrete combinatorics with continuous calculus.

A common trap is relying on Stirling's approximation ( $n! \sim \sqrt{2\pi n}(n/e)^n$ ), which correctly identifies the asymptotic behavior $M_n \sim 4^n/\sqrt{\pi n}$ but fails to establish rigorous, strict bounds for finite $n$ .

To bypass this, the student must construct a continuous analog using the Beta function $B(k+1, k+1) = \int_0^1 (x(1-x))^k dx$ .

By evaluating this integral for integer and half-integer values, and applying algebraic inequalities to the integrand $x(1-x) \le 1/4$ , the discrete binomial coefficient is rigorously bounded.

This highlights the JEE Advanced theme of using definite integral inequalities to bound discrete sums or coefficients.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Binomial Theorem - Class 11
Concept	Greatest Binomial Coefficient
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free