# Proving $\dfrac{(n-1)^{2n-2}}{(n-2)^{n-2}}<n^n$ for $n\ge 3$

• MHB
• anemone
In summary, the purpose of proving the inequality $\dfrac{(n-1)^{2n-2}}{(n-2)^{n-2}}<n^n$ for $n\ge 3$ is to establish the superiority of n as a base compared to (n-1) or (n-2) in certain mathematical equations or problems. The exponents (2n-2) and (n-2) determine the rate of growth for each expression, with the expression on the left side growing at a slower rate as n increases. The inequality can be proven using mathematical induction and is necessary to restrict the values of n to be greater than or equal to 3. This inequality has potential applications in mathematical
anemone
Gold Member
MHB
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Prove that $\dfrac{(n-1)^{2n-2}}{(n-2)^{n-2}}<n^n$ for integer $n\ge 3$.

Recall Bernoulli's inequality, $(1+x)^t>1+tx$ when $x>-1,\,x\ne 0$ and $t\ge 1$. For $m>1$, we have

\begin{align*}\left(\dfrac{m+1}{m}\right)^{m+1}\left(\dfrac{m-1}{m}\right)^{m-1}&=\left(1+\dfrac{1}{m}\right)^2\left(1-\dfrac{1}{m^2}\right)^{m-1}\\& >\left(1+\dfrac{1}{m}\right)^2\left(1-\dfrac{m-1}{m^2}\right)\\&=\left(\dfrac{m^3+1}{m^3}\right)\left(\dfrac{m+1}{m}\right)\\& >1\end{align*}

Hence, $(m+1)^{m+1}>m^{2m}(m-1)^{-(m-1)}$. Setting $m=n-1$ yields the desired result.