If `a_(1), a_(2), a_(3), …..a_(n)` is an arithmetic progression with common difference d. Prove that `tan[tan^(-1)((d)/(1+a_(1)a_(2)))+tan^(-1)((d)/(1+a_(2)a_(3)))+…+tan^(-1)((d)/(1+a_(n)a_(n-1)))]=(a_(n)-a_(1))/(1+a_(1)a_(n))`

If a_(1), a_(2), a_(3), ......... , are in A.P. with common difference 5 and if a_(i)a_(j) != - 1 for i ,j = 1, 2, .... , n , then tan^(1)(5/(1 + a_(1) a_(2))) + tan ^(1)(5/(1 + a_(2)a_(3))) + ...... + tan^(-1)(5/(1 + a_(n - 1)a_(n))) is equal to

If A_(1), A_(2),..,A_(n) are any n events, then

If the sequence a_(1),a_(2),a_(3),…,a_(n) is an A.P., then prove that a_(1)^(2)-a_(2)^(2)+a_(3)^(2)-a_(4)^(2)+…+a_(2n-1)^(2)-a_(2n)^(2)=n/(2n-1)(a_(1)^(2)-a_(2n)^(2))

If a_(1), a_(2), ……., a_(n) are in A.P. with common differece d != 0 , then (sin d) [sec a_(1) sec a_(2) + sec a_(2) sec a_(3) + .... + sec a_(n - 1) sec a_(n)] is equal to

If a_(1), a_(2), a_(3), a_(4), a_(5) and a_(6) are six arithmetic means between 3 and 31, then a_(6) - a_(5) and a_(1) + a_(6) are respectively =

If a_(1),a_(2),a_(3),…. are in A.P., then a_(p),a_(q),q_(r) are in A.P. if p,q,r are in

Let a_(1),a_(2),a_(3), . . . be a harmonic progression with a_(1)=5anda_(20)=25 . The least positive integer n for which a_(n)lt0 , is