If `a_1,a_2.......a_n` are `n` positive real numbers such that `a_1.a_2......a_n=1.` show that `(1+a_1)(1+a_2)......(1+a_n) >= 2^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_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

Let a_1,a_2,.........a_n be real numbers such that sqrt(a_1)+sqrt(a_2-1)+sqrt(a_3-2)++sqrt(a_n-(n-1))=1/2(a_1+a_2+.......+a_n)-(n(n-3)/4 then find the value of sum_(i=1)^100 a_i

Let a_1,a_2,.........a_n be real numbers such that sqrt(a_1)+sqrt(a_2-1)+sqrt(a_3-2)++sqrt(a_n-(n-1))=1/2(a_1+a_2+.......+a_n)-(n(n-3)/4 then find the value of sum_(i=1)^100 a_i

Let a_1,a_2,.........a_n be real numbers such that sqrt(a_1)+sqrt(a_2-1)+sqrt(a_3-2)++sqrt(a_n-(n-1))=1/2(a_1+a_2+.......+a_n)-(n(n-3)/4 then find the value of sum_(i=1)^100 a_i

If a_1,a_2,a_3,....a_n are positive real numbers whose product is a fixed number c, then the minimum value of a_1+a_2+.......a_(n-1)+2a_n is

If a_1,a_2,a_3,....a_n are positive real numbers whose product is a fixed number c, then the minimum value of a_1+a_2+....+a_(n-1)+3a_n is

If a_1,a_2,a_3,....a_n are positive real numbers whose product is a fixed number c, then the minimum value of a_1+a_2+....+a_(n-1)+3a_n is

If a_1,a_2,a_3,....a_n are positive real numbers whose product is a fixed number c, then the minimum value of a_1+a_2+....+a_(n-1)+3a_n is