JEE Advance - Mathematics (1987 - No. 15)
Prove by mathematical induction that $$ - 5 - {{\left( {2n} \right)!} \over {{2^{2n}}{{\left( {n!} \right)}^2}}} \le {1 \over {{{\left( {3n + 1} \right)}^{1/2}}}}$$ for all positive integers $$n$$.
Base Case: Show the inequality holds for n = 1.
Inductive Hypothesis: Assume the inequality holds for some positive integer k.
Inductive Step: Prove the inequality holds for k + 1, assuming it holds for k.
Consider the function f(n) = - 5 - (2n)! / (2^(2n) * (n!)^2) - 1 / (3n + 1)^(1/2) and show f(n) <= 0 for all positive integers n.
The given statement cannot be proven using mathematical induction.
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