If `a_1,a_2,a_3..........,a_n` is a GP of nonzero non-negative numers, then `loga_1,loga_2,loga_3,............,loga_n` is an AP and vice versa.

If a_1,a_2,a_3,.....a_n.... are in G.P. then the determinant Delta=|[loga_n,loga_(n+1),loga_(n+2)],[loga_(n+3),loga_(n+4),loga_(n+5)],[loga_(n+6),loga_(n+7),loga_(n+8)]| is equal to- (A) -2 (B) 1 (C) -1 (D) 0

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

If a_1+a_2+a_3+......+a_n=1 AA a_1 > 0, i=1,2,3,......,n, then find the maximum value of a_1 a_2 a_3 a_4 a_5......a_n.

If a_1,a_2,a_3,….,a_n are in G.P. are in a_i > 0 for each I, then the determinant Delta=|{:(loga_n,log a_(n+2),log a_(n+4)),(log a_(n+6),log a_(n+8), log a_(n+10)),(log a_(n+12), log a_(n+14), loga_(n+16)):}|

"If "a_1,a_2,a_3,.....,a_n" are in AP, prove that "a_(1)+a_(n)=a_(r)+a_(n-r+1)""

If a_1, a_2, a_3..... a_n in R^+ and a_1.a_2.a_3.........a_n = 1, then minimum value of (1+a_1 + a_1^2) (1 + a_2 + a_2^2)(1 + a_3 + a_3^2)........(1+ a_n + a_n^2) is equal to :-

If a_1,a_2,a_3,.....,a_n are in AP where a_i ne kpi for all i , prove that cosec a_1* cosec a_2+ cosec a_2* cosec a_3+...+ cosec a_(n-1)* cosec a_n=(cota_1-cota_n)/(sin(a_2-a_1)) .