Question

Let \(\mathrm{I}(\mathrm{n})=\mathrm{n}^{\mathrm{n}}, \mathrm{J}(\mathrm{n})=1.3 .5 \ldots \ldots(2 \mathrm{n}-1)\) for all \((\mathrm{n}>1), \mathrm{n} \in \mathrm{N}\), then
(A) \(\mathrm{I}(\mathrm{n})>\mathrm{J}(\mathrm{n})\)
(B) \(I(n)<J(n)\)
(C) \(\mathrm{I}(\mathrm{n})=\mathrm{J}(\mathrm{n})\)
(D) \(I(n)=\frac{1}{2} J(n)\)

Answer 1

Ans : (A)
Hint : AM \(\geq\) GM
\(\frac{1+3+5+7+\ldots+(2 n-1)}{n}>(J(n))^{\frac{1}{n}}, \quad \frac{n^{2}}{n}>(J(n))^{\frac{1}{n}}, \quad n^{n}>J(n), \quad I(n)>J(n)\)

Answer 2

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Answer 3

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