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.

The denominator is always the hypotenuse and the sequence looks to be strictly increasing. — Andrew Chin, Dec 03 '19 at 16:27
@TheDemonix_Hermit Well-spotted! So they appear to be the sines of the acute angles of right triangles with rational sides, in increasing order. — saulspatz, Dec 03 '19 at 16:27

Pspl · Accepted Answer · 2019-12-03T16:56:20.540

3

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

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

edited Dec 03 '19 at 16:56

answered Dec 03 '19 at 16:33

Pspl

does not work..... – James Dec 03 '19 at 16:54
It works for your set... – Pspl Dec 03 '19 at 16:55
check a_3 ..... – James Dec 03 '19 at 17:02
1

$a_3=8/10=4/5$. Similarly, $a_5=24/26=12/13$ etc – almagest Dec 03 '19 at 17:07
@almagest, thanks :) – Pspl Dec 03 '19 at 17:24

1 Answers1