It can be answered by using the concept of partial fraction
$\dfrac{1}{x\left(x+1\right)} =\dfrac{A}{x} + \dfrac{B}{\left(x+1\right)}$
solving this we will get $A=1$ and $B=-1$
So, this will form a sequence in which $2$ terms will remain $\left(1-\dfrac{1}{100}\right)$ and we will get $\dfrac{99}{100}=0.99$ as answer.