Question

If $a_1,a_2,.......,a_n$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

• A. $n(2c{)}^{\frac{1}{n}}$
• B. $(n+1)c{}^{\frac{1}{n}}$
• C. $2n{c}^{\frac{1}{n}}$
• D. $(n+1)(2c{)}^{\frac{1}{n}}$

Answer 1

The correct option is A $n(2c{)}^{\frac{1}{n}}$

Since, $A.M.\ge G.M.,wehave\frac{a_1+a_2+....+a_{n-1}+2a_n}{n}\ge {\left(a_1{a}_2....a_{n-1}2a_n\right)}^{\frac{1}{n}}\Rightarrow \frac{a_1+a_2+....+a_{n-1}+2a_n}{n}\ge {\left(2c\right)}^{\frac{1}{n}}\Rightarrow a_1+a_2+....+a_{n-1}+2a_n\ge n{\left(2c\right)}^{\frac{1}{n}}$