If `a_1,a_2,a_3,…………..a_n` are in A.P. whose common difference is d, show tht `sum_2^ntan^-1 d/(1+a_(n-1)a_n)= tan^-1 ((a_n-a_1)/(1+a_na_1))`

If a_1, a_2,a_3, ,a_n is an A.P. with common difference d , then prove that "tan"[tan^(-1)(d/(1+a_1a_2))+tan^(-1)(d/(1+a_2a_3))+tan^(-1)(d/(1+a_(n-1)a_n))]=((n-1)d)/(1+a_1a_n)

If a_1, a_2,a_3, ,a_n is an A.P. with common difference d , then prove that "tan"[tan^(-1)(d/(1+a_1a_2))+tan^(-1)(d/(1+a_2a_3))+tan^(-1)(d/(11+a_(n-1)a_n))]=((n-1)d)/(1+a_1a_n)

If a_1,a_2,a_3,……..a_n, a_(n+1),…….. be A.P. whose common difference is d and S_1=a_1+a_2+……..+a_n, S_2=a_(n+1)+………..+a_(2n), S_3=a_(2n+1)+…………+a_(3n) etc show that S_1,S_2,S_3, S_4………… are in A.P. whose common difference is n^2d .

If a_1,a_2,a_3,…….a_n are in Arithmetic Progression, whose common difference is an integer such that a_1=1,a_n=300 and n in[15,50] then (S_(n-4),a_(n-4)) is

If a_1,a_2,a_3, ,a_n are an A.P. of non-zero terms, prove that 1/(a_1a_2)+1/(a_2a_3)++1/(a_(n-1)a_n)= (n-1)/(a_1a_n)

let a_1,a_2,...a_n be in A.P. wth common difference pi/6 . if seca_1seca_2+seca_2seca_3+....seca_(n-1)seca_n = k(tana_n-tana_1) Find the value of k

If a_1,a_2,……….,a_n are in H.P. then expression a1a2+a2a3+......+a_n-1an

If a_1,a_2,a_3, ,a_n are in A.P. with common difference d(w h e r ed!=0) , then the sum of series. sind (coseca_1 coseca_2+coseca_2 coseca_3 +....+ coseca_(n-1) cos e ca_n) is equal to cot a_1-cot a_n

If a_1,a_2,a_3, ,a_n are in A.P., where a_i >0 for all i , show that 1/(sqrt(a_1)+sqrt(a_2))+1/(sqrt(a_1)+sqrt(a_3))++1/(sqrt(a_(n-1))+sqrt(a_n))=(n-1)/(sqrt(a_1)+sqrt(a_n))dot

If a_1,a_2,a_3, ,a_n are in A.P., where a_i >0 for all i , show that 1/(sqrt(a_1)+sqrt(a_2))+1/(sqrt(a_1)+sqrt(a_3))++1/(sqrt(a_(n-1))+sqrt(a_n))=(n-1)/(sqrt(a_1)+sqrt(a_n))dot