If `A_1, A_2,.....A_n` are independent events associated with a random experiment; then `P(A_1 nn A_2 nn .......nn A_n) = P(A_1)P(A_2).......P(A_n)`

If A_1, A_2, …, A_n are n independent events, such that P(A_i)=(1)/(i+1), i=1, 2,…, n , then the probability that none of A_1, A_2, …, A_n occur, is

.Let a_1, a_2,............ be positive real numbers in geometric progression. For each n, let A_n G_n, H_n, be respectively the arithmetic mean, geometric mean & harmonic mean of a_1,a_2..........a_n. Find an expression ,for the geometric mean of G_1,G_2,........G_n in terms of A_1, A_2,........ ,A_n, H_1, H_2,........,H_n.

Let A_1, A_2........A_7 be a polygon and a_1........a_7 be the complex numbers representing vertices A_1,A_2........A_7. if, |a_1|=|a_2|=.............|a_7|=R, then sum_(1 le j le 7) |a_1+a_2|^2

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_n are n distinct odd natural numbers not divisible by any prime greater than 5, then prove that (1)/(a_1)+(1)/(a_2)+…..+(1)/(a_n) lt 2 .

"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=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 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)) .