Question

Prove that 2.4.6 ........... (2n) < (n + 1)ⁿ.

Answer 1

Applying AM - GM, inequality, we have

\(\frac{2+4+6+....+2n}{n}>(2.4.6....2n)^{\frac1n}\)

⇒ \(\frac1n\big[\frac{n}2(3+2n)\big](2.4.6.....2n)^{\frac1n}\) [Sum of an AP [First term Ap = \(\frac{n}2\)Last term]]

⇒ (n + 1) > (2.4.6......2n)^\(\frac1n\) ⇒  (2.4.6......2n)^\(\frac1n\) < (n + 1) ⇒ (2.4.6......2n) < (n + 1)ⁿ.