Here is an inequality (let's call it - "$A(n)$") that has to be proved: $$ (n+1)^n < n^{n+1} \text{ for } n\ge3. $$
I'll skip the first two steps of induction and move right to the induction step (step #3) in order not to waste time.
- Assume that $n = k+1$. I get: $$ (k+2)^{k+1} < (k+1)^{k+2} $$ From here: $$ (k+2)^k(k+2) < (k+1)^k(k+1)^2 $$ and that is where i'm stuck.
I've found, that the inequality $A(n)$ (by dividing it by $n^n$) can be expressed as: $$ (1+1/n)^n < n. $$ Here again I substitute $n = k+1$ and get: $$ (1+1/(k+1))^{k+1} < k+1. $$ Stuck here as well...
On the forum I've found ways to prove the inequality by Binomial theorem and other approaches, but how to prove this with math induction ? It would be great if answers included comments on every step, that is done to find the solution...
