Question

The sequence $\left\{x_{k}\right\}$ is defined by $x_{k+1}=x_{k}^{2}+x_{k}$ and $x_{1}=\frac{1}{2} .$ Then $\left[\frac{1}{x_{1}+1}+\frac{1}{x_{2}+1}+\ldots+\frac{1}{x_{100}+1}\right]$ (where [.] denotes the greatest integer function) is equal to

• A. 0
• B. 2
• C. 4
• D. 1

Answer 1

Answer: (D)

$\frac{1}{x_{k+1}}=\frac{1}{x_{k}\left(x_{k}+1\right)}=\frac{1}{x_{k}}-\frac{1}{x_{k}+1}$
$\Rightarrow \frac{1}{x_{k+1}}=\frac{1}{x_{k}}-\frac{1}{x_{k-1}}$
$\therefore \frac{1}{x_{1}+1}+\frac{1}{x_{2}+1}+\ldots+\frac{1}{x_{100}+1}=\frac{1}{x_{1}}-\frac{1}{x_{101}}$
$\therefore $ As $0<\,\frac{1}{x_{101}}<\,1 $
$\therefore \left[\frac{1}{x_{1}+1}+\frac{1}{x_{2}+1}+\ldots+\frac{1}{x_{100}+1}\right]=1$