Sum of $n$ terms of the series $\frac 1{(1 + x)(1 +2x)} + \frac 1{(1 +2x)(1 +3x)} + \ldots$ is equal to?

Question

My try : By using Vn Method

But the answer is $\frac n{(1+x)(1+ (n+1)x)}$
Please tell me where I am wrong or what is the correct method to solve it.
Thanks

Answer 1

I am sorry but, being almost blind, I am unable to read your photograph.

Considering $$a_n=\frac n{(1+x)(1+ (n+1)x)}$$ use patrial fraction decomposition to get $$a_n=\frac{n+1}{(n+1) x+1}-\frac{n}{n x+1}$$ which beautifully telescopes.

I am sure that you can take it from here.