Very simple example:
$1+ \frac{1}{4} + \frac{1}{9} + ...+ \frac{1}{n^2} < 2$
cannot be proved by induction for obvious reasons, but
$1+ \frac{1}{4} + \frac{1}{9} + ...+ \frac{1}{n^2} < 2-\frac{1}{n}$
is an easy induction problem.
[Edit]: Forgot to add this, while the example is very simple, I like the fact that it is easy to understand (before solving the problem) why induction cannot work in the first example and why it could work in the stronger case.
