Mathematics - Sequence & Series Question with Solution | TestHub

$t_r=\sqrt{1+\frac{1}{r^2}+\frac{1}{(r+1{)}^2}}$

$$\begin{array}{rl}& =\sqrt{\frac{r^2+(r+1{)}^2+r^2(r+1{)}^2}{r^2(r+1{)}^2}}\\ & =\sqrt{\frac{2r^2+2r+1+r^2\left(r^2+2r+1\right)}{r^2(r+1{)}^2}}\\ & =\sqrt{\frac{r^4+2r^3+3r^2+2r+1}{r^2(r+1{)}^2}}\\ & =\frac{r^2+r+1}{r(r+1)}=\frac{1}{r(r+1)}+1\\ & =1+\frac{1}{r}-\frac{1}{r+1}\end{array}$$

Given $S=2000-\frac{1}{2000}$, we have $|2000(S-2000)|=1$.