Find the pattern of a series

Question

I am given the series $$\frac{1}{6} + \frac{1\cdot 8}{6\cdot 10} + \frac{1\cdot 8\cdot 15}{6\cdot 10\cdot 14}$$

I am asked to find a formula for this series $a_n$

So I have found that the last fraction multiplied at $nth$ term is $\frac{(1+7(n-1))}{(6+4(n-1))}$

But I am stuck, unsure how to proceed after this.

Answer 1

Try recursion.

You know that each term (after the first term, of course) is simply the previous term multiplied by a varying constant that relates to $n$.

So try something like:

$$a_1 = \frac{1}{6} $$ and $$a_n = a_{n-1} \cdot \frac{1+7(n-1)}{6+4(n-1)}.$$