Inequalities 🥲 (1 Viewer)

Hello_World2 · Mar 27, 2023

How do I proceed further after the n=k+1 step, I'm completely stuck. Pls help thanks !!

tywebb · Mar 28, 2023

You may proceed to simplify the induction hypothesis to make it easier to do the induction step.

So your induction hypothesis assume $k^{k+1}>(k+1)^k$ is equivalent to $\textstyle(1+\frac{1}{k})^k<k$ . You can see why by dividing by $k^k$ .

Now in order to do the induction step, one only needs to prove $\textstyle(1+\frac{1}{k+1})^{k+1}<k+1$

$\begin{aligned}\textstyle(1+\frac{1}{k+1})^{k+1}&<\textstyle(1+\frac{1}{k})^{k+1}\text{ since }\frac{1}{k+1}<\frac{1}{k}\\&=\textstyle(1+\frac{1}{k})^k(1+\frac{1}{k})\\&<\textstyle k(1+\frac{1}{k})\text{ from the induction hypothesis}\\&=k+1\end{aligned}$

$\text{Hence }\textstyle(1+\frac{1}{k+1})^{k+1}<k+1\ \therefore\ (k+1)^{k+2}>(k+2)^{k+1}$

Inequalities 🥲 (1 Viewer)

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