If $a_{1}, a_{2}, \ldots \ldots . . ., a_{n}$ are positive real numbers whose product is a fixed num

Question

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

Professor PC Thomas Classes · Answer

(a) : $\frac{\left(a_{1}+a_{2}+\ldots .+a_{n-1}+2 a_{n}ight)}{n}$ $$ \begin{gathered} \geq\left(a_{1} a_{2} \ldots a_{n-1} 2 a_{n}ight)^{1 / n} \ a_{1}+a_{2}+\ldots \ldots+a_{n-1}+2 a_{n} \geq n\left(a_{1} a_{2} \ldots \ldots a_{n-1}ight. \ \left.\left.2 a_{n}ight)^{1 / n}+\ldots . .+ 	ext { Using A.M. } \geq 	ext { G.M. }ight] \ \Rightarrow a_{1}+a_{2}+a_{3}+\ldots+a_{n-1}+2 a_{n} \geq n(2 c)^{1 / n} . \end{gathered} $$