Finding an explicit formula for a sequence

Question

Suppose we have the sequence

$$ \left(0, {3\over5}, {4\over5}, {15\over17}, {12\over13},{35\over37},\ldots\right).$$

Is it possible to find an explicit formula $a_n$? I cant seem to find one.

Answer 1

If you know the rule to generate your sequence, you may prove by induction that:

$$a_n=\frac{n^2-1}{n^2+1}, n\ge 1$$