If {a_1}, {a_2}, ..., {a_n} are positive real | vedclass

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(A) By the $AM-GM$ inequality,for $n$ positive real numbers,the arithmetic mean is greater than or equal to the geometric mean.Consider the $n$ number

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

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(A) By the $AM-GM$ inequality,for $n$ positive real numbers,the arithmetic mean is greater than or equal to the geometric mean.Consider the $n$ numbers: ${a_1}, {a_2}, ..., {a_{n-1}}, 2{a_n}$.Their arithmetic mean is $\frac{{a_1} + {a_2} + ... + {a_{n-1}} + 2{a_n}}{n}$.Their geometric mean is $({a_1} \cdot {a_2} \cdot ... \cdot {a_{n-1}} \cdot 2{a_n})^{1/n} = (2 \cdot {a_1} \cdot {a_2} \cdot ... \cdot {a_n})^{1/n} = (2c)^{1/n}$.Applying $AM \ge GM$:$\frac{{a_1} + {a_2} + ... + {a_{n-1}} + 2{a_n}}{n} \ge (2c)^{1/n}$.Therefore,the minimum value of ${a_1} + {a_2} + ... + {a_{n-1}} + 2{a_n}$ is $n(2c)^{1/n}$.

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

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