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

Explanation

`therefore AM ge GM`
` therefore (a_(1)+a_(2)+"......."+a_(n-1)+3a_(n))/(n)ge (a_(1)a_(2)"...."a_(n-1)3a_(n))^((1)/(n))=(3c)^((1)/(n))`
` implies a_(1)+a_(2)+"......."+a_(n-1)+3a_(n)ge n (3c)^((1)/(n))`
Hence, the minimum value of ` a_(1)+a_(2)+"...."+a_(n-1)+3a_(n) " is " n(3c)^((1)/(n))`.