Sequences and Series question bank (Mathematics)

If $a_1, a_2, a_3, \ldots, a_n$ are in arithmetic progression with common difference d, then $\tan\left[\tan^{-1}\left(\dfrac{d}{1 + a_1 a_2}\right) + \tan^{-1}\left(\dfrac{d}{1 + a_2 a_3}\right) + \ldots + \tan^{-1}\left(\dfrac{d}{1 + a_{n-1} a_n}\right)\right] =$ ____

In a right-angled $\triangle ABC$, the measures of the angles are in an Arithmetic Progression (A.P.) If its smallest side is $4$ units, then the area of $\triangle ABC$ is.....