Answer
$a_{n}=n^{2}-1,\quad n=1,2,3...$
Work Step by Step
The differences between terms are $3,5,7,9,...$
These are not constant.
Observe the sequence
$1,4,9,16,25,...$
(the sequence of squares of natural numbers).
We see that the differences between terms are also $3,5,7,9$...,
so we seek a relation between these two sequences.
We find it by observing the pattern:
$ 1=1-1\quad$ = $n^{2}-1$ when $n=1,$
$ 3=4-1\quad$ = $n^{2}-1$ when $n=2,$
$ 8=9-1\quad$ = $n^{2}-1$ when $n=3,$
$ 15=16-1\quad$ = $n^{2}-1$ when $n=4$
$ 24=25-1\quad$ = $n^{2}-1$ when $n=5$
$....$
$a_{n}=n^{2}-1,\quad n=1,2,3...$
