Question

For each positive integer n, prove that (1+1/n)^{n}<(1+1/n+1)^{n}

Answer 1

What you've written isn't true.

But what is true is that for all positive integers n,
\[ \left(1 + \frac{1}{n}\right)^n < \left(1 + \frac{1}{n+1}\right)^{n+1} \]