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Let $\mathrm{I}(\mathrm{n})=\mathrm{n}^{\mathrm{n}}, \mathrm{J}(\mathrm{n})=1.3 .5 \ldots . .(2 \mathrm{n}-1)$ for all $(\mathrm{n}>1), \mathrm{n} \in \mathrm{N}$, then
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Verified Answer
The correct answer is:
$\mathrm{I}(\mathrm{n})>\mathrm{J}(\mathrm{n})$
Hint:
$\mathrm{AM} \geq \mathrm{GM}$
$\frac{1+3+5+7+\ldots+(2 n-1)}{\mathrm{n}}>(\mathrm{J}(\mathrm{n}))^{\frac{1}{n}}, \quad \frac{\mathrm{n}^{2}}{\mathrm{n}}>(\mathrm{J}(\mathrm{n}))^{\frac{1}{n}}, \quad \mathrm{n}^{\mathrm{n}}>\mathrm{J}(\mathrm{n}), \mathrm{I}(\mathrm{n})>\mathrm{J}(\mathrm{n})$
$\mathrm{AM} \geq \mathrm{GM}$
$\frac{1+3+5+7+\ldots+(2 n-1)}{\mathrm{n}}>(\mathrm{J}(\mathrm{n}))^{\frac{1}{n}}, \quad \frac{\mathrm{n}^{2}}{\mathrm{n}}>(\mathrm{J}(\mathrm{n}))^{\frac{1}{n}}, \quad \mathrm{n}^{\mathrm{n}}>\mathrm{J}(\mathrm{n}), \mathrm{I}(\mathrm{n})>\mathrm{J}(\mathrm{n})$
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