Summation this Summer!

Algebra Level 3

$\large \dfrac{1}{(1+x)(1+2x)}+\dfrac{1}{(1+2x)(1+3x)}+\dfrac{1}{(1+3x)(1+4x)} + \cdots$

Find the closed form of the sum of the series above up to $n$ terms.

\frac{n}{(1+x)[1+(n^2+1)x]}

\frac{n}{(1+x)[1+(n+1)x]}

\frac{n+1}{(1+x)[1+(n+1)x]}

\frac{n^2}{(1+x)[1+(n+1)x]}