Question

If
\[ \frac{4^n}{n+1} < \frac{(2n)!}{(n!)^2}, \]
then \(P(n)\) is true for

• A. \(n\ge1\)
• B. \(n>0\)
• C. \(n<0\)
• D. n\ge2

Hint:

Test boundary values to determine validity range.

Answer 1

The Correct Option is D

Checking small values:

\(n = 1: \frac{4}{2} = 2 = \frac{2!}{1! \, 1!} \) (not <)

\(n = 2: \frac{16}{3} < \frac{24}{4} \)

Thus the inequality holds for \(n \ge 2\).