Prove by induction that (2n)! 2²ⁿ(n!)² ≤ 1 √(3n+1) for all positive integers n.

Answer: Proved. Base n=1: 12≤ 12. The step requires (2n+1)/(2n+2)≤√((3n+1)/(3n+4)), which holds since cross-multiplying gives (2n+1)²(3n+4)≤(2n+2)²(3n+1). JEE Advanced 1987 · Binomial Theorem — verified solution by the Vidaara Team.

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[JEE Advanced 1987] prove by induction that 2n 2 2n n 2 1 sqrt 3n 1 for

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Prove by induction that $\dfrac{(2n)!}{2^{2n}(n!)^2}\le\dfrac1{\sqrt{3n+1}}$ for all positive integers $n$.

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VAVidaara Admin ✓ Vidaara Team ✓ Accepted · 2d ago ▲ 0

Answer: Proved.

Base $n=1$: $\frac12\le\frac12$. The step requires $\frac{2n+1}{2n+2}\le\sqrt{\frac{3n+1}{3n+4}}$, which holds since cross-multiplying gives $(2n+1)^2(3n+4)\le(2n+2)^2(3n+1)$.

JEE Advanced 1987 · Binomial Theorem — verified solution by the Vidaara Team.

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