Question

If $a_1,a_2,...,a_n$ are positive real numbers whose product is a fixed number c, the minimum value of $a_1+a_2+...+a_{n-1}+2a_n$ is

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

Answer 1

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

Given that, $a_1{a}_2...a_{n-1}{a}_n=c$
Since, $A.M\ge G.M$
Therefore, $\frac{a_1+a_2+...+a_{n-1}+2a_n}{n}\ge {(a_1{a}_2...a_{n-1}2a_n)}^{1/n}$
$\Rightarrow a_1+a_2+...+a_{n-1}+2a_n\ge {n(2a_1{a}_2...a_{n-1}{a}_n)}^{1/n}$
$\Rightarrow a_1+a_2+...+a_{n-1}+2a_n\ge {n\left(2c\right)}^{1/n}$
Ans: A