A sequence $x_n$ is defined as $x_{k+1}=x_k^2+x_k$ and $x_1=\frac{1}{2}$ Let:
$$S_n=\left[\frac{1}{x_1+1}+\frac{1}{x_2+1}+\frac{1}{x_3+1}+\cdots+\frac{1}{x_n+1}\right]$$
where $[\cdot]$ denotes greatest integer function. What is value of $S_{100}$?
By manipulating the definition of series i deduced that $x_k+1=\frac{x_{k+1}}{x_k}$. So the series becomes $\left[\displaystyle\sum_{k=1}^n\frac{x_k}{x_{k+1}}\right]$. But what to do after that?
Note that $(x_n)_n$ is increasing and cannot be bounded so $\lim\limits_{n\to\infty}x_n=+\infty$. Also, $$\frac{1}{x_{n+1}}=\frac{1}{x_n}-\frac{1}{x_n+1}$$ hence $$\sum_{k=1}^n\frac{1}{1+x_k}=\frac{1}{x_1}-\frac{1}{x_{n+1}}=2-\frac{1}{x_{n+1}}$$ Now, $x_1=3/4$ and $x_2>1$ hence for $n\ge1$ we have $x_{n+1}>1$, and consequently $$\forall\,n\ge1,\qquad 2>2-\frac{1}{x_{n+1}}>1$$ Thus $\left[2-\frac{1}{x_{n+1}}\right]=1$ for every $n\ge1$. In particular, $S_{100}=1$.
