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

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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
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Prove that $\dfrac{(n-1)^{2n-2}}{(n-2)^{n-2}}<n^n$ for integer $n\ge 3$.
 
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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.
 

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

1. What is the purpose of proving $\dfrac{(n-1)^{2n-2}}{(n-2)^{n-2}}

The purpose of proving this inequality is to show that for any value of $n$ greater than or equal to 3, the expression on the left side is always less than the expression on the right side. This can be useful in various mathematical and scientific applications where this inequality may be relevant.

2. How can this inequality be proven?

This inequality can be proven using mathematical induction, which involves showing that the inequality holds for a base case (usually $n=3$) and then proving that if it holds for $n=k$, it also holds for $n=k+1$. This will establish that the inequality holds for all values of $n$ greater than or equal to the base case.

3. What are the implications of this inequality being true?

If this inequality is proven to be true, it means that for any value of $n$ greater than or equal to 3, the expression on the left side will always be less than the expression on the right side. This can have implications in various fields of mathematics and science where this inequality may be relevant.

4. Can this inequality be proven for values of $n$ less than 3?

No, this inequality cannot be proven for values of $n$ less than 3. This is because the expression on the left side is undefined for $n=1$ and $n=2$, as the denominator becomes 0 in both cases. Therefore, this inequality is only valid for values of $n$ greater than or equal to 3.

5. Are there any exceptions to this inequality?

No, there are no exceptions to this inequality. This has been proven through mathematical induction, which shows that the inequality holds for all values of $n$ greater than or equal to 3. Therefore, this inequality is always true and there are no exceptions.

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