Find the formula for a given sequence 3/6, 8/25, 15/62, 24/123, 35/214

Find the formula for a given sequence 3/6, 8/25, 15/62, 24/123, 35/214.

If we add to the numerators +1, we will have the sequence 4, 9, 16, 25, 36,... These are squared integers, therefore, the numerators satisfy the formula $n^2-1$ , ($n\geq 2).$

Now let us flip the fractions: 6/3=2, 25/8=3+1/8, 62/15=4+2/15, 123/24=5+3/24, 214/35=6+4/35. The formula of common term can be easily guessed: n+(n-2)/(n2-1). Returning to the original fractions, we have

$\frac{1}{n+(n-2)/(n^2-1)}=\frac{n^2-1}{n(n^2-1)+(n-2)}=\frac{n^2-1}{n^3-2}$

Answer. $(n^2-1)/(n^3-2)$ , ($n\geq 2).$