Question

If $\tan ({\tan }^{-1}\frac{1}{3}+{\tan }^{-1}\frac{1}{7}+{\tan }^{-1}\frac{1}{13}+\cdots +{\tan }^{-1}\frac{1}{307})$ $=\frac{p}{q},$ where $p,q\in N,$ then the value of $\left[\frac{q}{p}\right]$ is (where $\left[y\right]$ represents greatest integer function of $y$.)

Answer 1

$\tan ({\tan }^{-1}\frac{1}{1+1\cdot 2}+{\tan }^{-1}\frac{1}{1+2\cdot 3}+{\tan }^{-1}\frac{1}{1+3\cdot 4}+...+{\tan }^{-1}\frac{1}{1+17\cdot 18})$
$=\tan ({\tan }^{-1}2-{\tan }^{-1}1+{\tan }^{-1}3-{\tan }^{-1}2+...+{\tan }^{-1}18-{\tan }^{-1}17)$
$=\tan ({\tan }^{-1}18-{\tan }^{-1}1)$
$=\tan \left({\tan }^{-1}\frac{18-1}{1+18}\right)=\frac{17}{19}=\frac{p}{q}$
$\therefore \left[\frac{q}{p}\right]=1$