Question Number 172560 by mr W last updated on 28/Jun/22
$${prove}\:{for}\:{n}\in{N}\:{and}\:{n}>\mathrm{1} \\ $$$$\left(\frac{{n}+\mathrm{1}}{\mathrm{3}}\right)^{{n}} <{n}!<\left(\frac{{n}+\mathrm{1}}{\mathrm{2}}\right)^{{n}} \\ $$
Commented by mr W last updated on 07/Sep/22
$${see}\:{also}\:{Q}\mathrm{173815} \\ $$
Answered by Jamshidbek last updated on 28/Jun/22
$$\mathrm{Hint}:\:\mathrm{Mathematic}\:\mathrm{induction}\:\mathrm{method} \\ $$
Answered by mnjuly1970 last updated on 28/Jun/22
$$\:\:\:\left({n}>\mathrm{1}\right)\::\frac{\:\mathrm{1}+\mathrm{2}+…+{n}}{{n}}\:>\sqrt[{{n}}]{\mathrm{1}.\mathrm{2}.\mathrm{3}…{n}} \\ $$$$\:\:\:\:\:\:\:\frac{{n}+\mathrm{1}}{\mathrm{2}}\:>\sqrt[{{n}}]{{n}!}\:\Rightarrow\:\left(\frac{{n}+\mathrm{1}}{\mathrm{2}}\:\right)^{\:{n}} >{n}! \\ $$