How to find Find $\left\lfloor\sum_1^{100}\frac{1}{x_n+1}\right\rfloor$ with $x_1 =\frac{1}{2}, x_{k+1} =x_k^2+x_k$.

How to find Find $\left\lfloor\sum_1^{100}\frac{1}{x_n+1}\right\rfloor$
with $x_1 =\frac{1}{2}, x_{k+1} =x_k^2+x_k$.

The sequence $\{x_n\}$ is defined by $x_1 =\frac{1}{2}, x_{k+1}
=x_k^2+x_k$. Find
$$\left\lfloor\frac{1}{x_1+1}+\frac{1}{x_2+1}+.........+\frac{1}{x_{100}+1}\right\rfloor$$
where $\left\lfloor\dots\right\rfloor$ is greatest integer function.
If we put the values of $k$ then we get the numerator part of the series
=2 and then taking 2 as common from the series how can we solve the rest
of the series..
How do we proceed in this case ... please suggest thanks..