If `a_1. a_2 ....... a_n` are positive and (n - 1) s = `a_1 + a_2 +.....+a_n` then prove that `(a_1 + a_2 +....+a_n)^n ge (n^2 - n)^n (s - a_1) (s - a_2)........(s - a_n)`

If a_1, a_2,...... ,a_n >0, then prove that (a_1)/(a_2)+(a_2)/(a_3)+(a_3)/(a_4)+.....+(a_(n-1))/(a_n)+(a_n)/(a_1)> n

If S=a_1+a_2+......+a_n,a_i in R^+ for i=1 to n, then prove that S/(S-a_1)+S/(S-a_2)+......+S/(S-a_n) ge n^2/(n-1), AA n ge 2

If a_1,a_2,.....a_n are positive real number whose product is a fixed number c, then the minimum value of a_1+a_2+......+a_(n-1)+a_n is

If a_1,a_2,.....a_n are in H.P., then the expression a_1a_2 + a_2a_3 + ... + a_(n-1)a_n is equal to

If the sequence a_1, a_2, a_3,....... a_n ,dot forms 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_3+......+a_n=1 AA a_i > 0, i=1,2,3,......,n , then find the maximum value of a_1 a_2 a_3 a_4 a_5......a_n .

Let a_1,a_2 ,a_3 ...... be an A.P. Prove that sum_(n=1)^(2m)(-1)^(n-1)a_n^2=m/(2m-1)(a_1^2-a_(2m)^2) .

Let a_1 , a_2 , a_3,.... be terms of an A.P. if (a_1 + a_2 +....+ a_p)/(a_1 + a_2 +...+ a_q) = p^2/q^2. p ne q then a_6/a_21 equals

Let vec r_1, vec r_2, vec r_3, .....vec r_n be the position vectors of points P_1,P_2, P_3 ,......P_n relative to the origin O . If the vector equation a_1 vec r_1+a_2 vec r_2+.....a_n vec r_n=0 hold, then a similar equation will also hold w.r.t. to any other origin provided a) a_1+a_2+......a_n=n b) a_1+a_2+....a_n=1 c) a_1+a_2+....a_n=0 d) a_1=a_2=a_3+a_n=0

If a ,a_1, a_2, a_3, a_(2n),b are in A.P. and a ,g_1,g_2,g_3, ,g_(2n),b . are in G.P. and h s the H.M. of aa n db , then prove that (a_1+a_(2n))/(g_1g_(2n))+(a_2+a_(2n-1))/(g_1g_(2n-1))++(a_n+a_(n+1))/(g_ng_(n+1))=(2n)/h