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_n))`

If a_1,a_2,a_3,...,a_n be in AP whose common difference is d then prove that sum_(i=1)^n a_ia_(i+1)=n{a_1^2+na_1d+(n^2-1)/3 d^2} .

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

Let a_1,a_2,a_3,.....,a_n be in GP whose common ratio is r. Show that sum_(k=1)^(n-1) 1/(a_k^2-a_(k+1)^2)=(1-r^(2(n-1)))/(a_1^2*r^(2(n-2))*(1-r^2)^2) .

If a_1, a_2, a_3,...a_n are in A.P with common difference d !=0 then the value of sind(coseca_1 coseca_2 +cosec a_2 cosec a_3+...+cosec a_(n-1) cosec a_n) will be

We know that, if a_1, a_2, ..., a_n are in H.P. then 1/a_1,1/a_2,.....,1/a_n are in A.P. and vice versa. If a_1, a_2, ..., a_n are in A.P. with common difference d, then for any b (>0), the numbers b^(a_1),b^(a_2),b^9a_3),........,b^(a_n) are in G.P. with common ratio b^d. If a_1, a_2, ..., a_n are positive and in G.P. with common ration, then for any base b (b> 0), log_b a_1 , log_b a_2,...., log_b a_n are in A.P. with common difference logor.If x, y, z are respectively the pth, qth and the rth terms of an A.P., as well as of a G.P., then x^(z-x),z^(x-y) is equal to