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: Sequence & Series (Mathematics)

(a) : Given $a_1 \cdot a_2 \cdot a_3 \ldots \ldots . a_n=c$ $\\ \Rightarrow a_1 \cdot a_2 \cdot a_3 \ldots \ldots . . a_{n-1}\left(2 a_n\right)=2 c\\$ We know that A.M. $\\ \geq$ G.M. A.M. of $a_1, a_2, \ldots \ldots, 2 a_n$ is $\frac{a_1+a_2+a_3+2 a_n}{n}\\$ G.M. of $a_1, a_2, a_3, \ldots \ldots, 2 a_n$ is $\left(a_1 a_2 a_3 \ldots \ldots \ldots .2 a_n\right)^{1 / n}\\$ $\text { ⇒ G.M. }=(2 c)^{1 / n}\\$ Now, A.M. $\geq$ G.M. $\Rightarrow a_1+a_2+a_3+\cdots \ldots+2 a_n \geq n(2 c)^{1 / n} \\$ ⇒ minimum value of $\left(a_1+a_2+a_3+\ldots . .+2 a_n\right)=n(2 c)^{1 / n}$

Explanation

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