If `a_1,\ a_2,\ a_3,\ ,\ a_n` are in arithmetic progression with common difference `d` , then evaluate the following expression: `tan{tan^(-1)(d/(1+a_1a_2))+tan^(-1)(d/(1+a_2a_3))+tan^(-1)(d/(1+a_3a_4))++tan^(-1)(d/(1+a_(n-1)a_n))}`

Explanation

`d=a_{2}-a_{1}=a_{3}-a_{2} ldots=a_{n}-a_{n-1}`

`tan {tan ^{-1}(frac{d}{1+a_{1} a_{2}})+tan ^{-1} frac{d}{1+a_{2} a_{3}}+ldots . tan ^{-1} frac{d}{1+a_{n-1} a_{n}}}`

`=tan {tan ^{-1}(frac{a_{2}-a_{1}}{1+a_{1} a_{2}})+tan ^{-1} frac{a_{3}-a_{2}}{1+a_{2} a_{3}}+ldots ldots . tan ^{-1} frac{a_{n}-a_{n-1}}{1+a_{n-1} a_{n}}}`

`=tan {a_{2}-a_{1}+a_{3}-a_{2}+a_{4}-a_{3}+ldots ldots a_{n}-a_{n-1}}`

`=tan {a_{n}-a_{1}}`

`=tan {a_{1}+(n-1) d-a_{1}}`

`=tan {(n-1) d}`