Question

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

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

Answer 1

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

Given that $a_1,a_2,...,a_n$ be positive real numbers such that $a_1\cdot a_2\cdot ...\cdot a_n=c$ $...\left(1\right)$

We know that, A.M $\ge$ G.M

$\Rightarrow \frac{a_1+a_2+...+a_{n-1}+2a_n}{n}\ge {(a_1{a}_2...a_{n-1}(2a_n\left)\right)}^{\frac{1}{n}}$

$\Rightarrow a_1+a_2+...+a_{n-1}+2a_n\ge {n\left(2c\right)}^{\frac{1}{n}}$ $...\left(from\right(1\left)\right)$

So, the minimum value of $a_1+a_2+...+a_{n-1}+2a_n$ is ${n\left(2c\right)}^{\frac{1}{n}}$.