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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.

: 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.