If a_1, a_2, dots, a_n are positive real numb | vedclass

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(A) By the Arithmetic Mean-Geometric Mean $(AM \geq GM)$ inequality,for positive real numbers $a_1, a_2, \dots, a_{n-1}, 2a_n$:$\frac{a_1 + a_2 + \dot

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$n(2c)^{1/n}$

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(A) By the Arithmetic Mean-Geometric Mean $(AM \geq GM)$ inequality,for positive real numbers $a_1, a_2, \dots, a_{n-1}, 2a_n$:$\frac{a_1 + a_2 + \dots + a_{n-1} + 2a_n}{n} \geq (a_1 \cdot a_2 \cdot \dots \cdot a_{n-1} \cdot 2a_n)^{1/n}$Given that the product $a_1 \cdot a_2 \cdot \dots \cdot a_n = c$,we can substitute this into the inequality:$\frac{a_1 + a_2 + \dots + a_{n-1} + 2a_n}{n} \geq (2 \cdot a_1 \cdot a_2 \cdot \dots \cdot a_n)^{1/n}$$\frac{a_1 + a_2 + \dots + a_{n-1} + 2a_n}{n} \geq (2c)^{1/n}$$a_1 + a_2 + \dots + a_{n-1} + 2a_n \geq n(2c)^{1/n}$Thus,the minimum value is $n(2c)^{1/n}$.

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

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