Prove that `(1+a_1)(1+a_2)...(1+a_n)>=2^n`

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 1, a_1,a_2,a_3 ,…, a_(n-1) are the nth roots of unity then prove that : (1-a_1)(1-a_2)(1-a_3)...(1-a_(n-1)) =n.

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_i>0,i=1,2,3,...n and (n-1)s=a_1+a_2+a_3+....+a_n , prove that a_1*a_2*a_3*....a_nge(n-1)^n*(s-a_1)(s-a_2)...(s-a_n) .

If S+a_1+a_2+a_3++a_n ,a_1 in R^+ for i=1ton , then prove that S/(S-a_1)+S/(S-a_2)++S/(S-a_n)geq(n^2)/(n-1),AAngeq2

If a_1, a_2, a_3 ….. a_n are in AP, then prove that frac(1)(a_1 a_2)+frac(1)(a_2 a_3)+frac(1)(a_3 a_4)+.....+frac(1)(a_(n-1) a_n)=frac(n-1)(a_1 a_n)

If a_i > 0 for i=1,2,…., n and a_1 a_2 … a_(n=1) , then minimum value of (1+a_1) (1+a_2) ….. (1+a_n) is :

If a_i > 0 for i=1,2,…., n and a_1 a_2 … a_(n=1) , then minimum value of (1+a_1) (1+a_2) ….. (1+a_n) is :

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