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Let $n>1$ be an integer. Show that
$$n !<\left(\frac{n+1}{2}\right)^{n}$$
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Applying AM-GM to the terms $x_{1}=1, x_{2}=2, \ldots, x_{n}=n$, we have
\begin{aligned} \frac{1+2+\cdots+n}{n} & \geq \sqrt[n]{1 \times 2 \times \cdots \times n} \\ \frac{\frac{n(n+1)}{2}}{n} & \geq \sqrt[n]{n !} \\ \left(\frac{n+1}{2}\right)^{n} & \geq n ! \end{aligned}
Since $n>1$, the terms $x_{i}$ are not all equal, and hence equality does not hold. As such, this shows that
$$\left(\frac{n+1}{2}\right)^{n}>n ! . \square$$
by Platinum (122,442 points)

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