If $a_{1}, a_{2}, a_{3}, . . . . ., a_{n}$ are positive real numbers whose product is a fixed number $c$ , then the minimum value of $a_{1} + a_{2} + a_{3} + . . . . . + a_{n - 1} + a_{n}$ is

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

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(n + 1) c^{1 / n}

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2 n c^{1 / n}

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

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Solution

The correct option is A $n {(c)}^{1 / n}$
We know that for positive quantities, $A M \geq G M$ .
Given, $(a_{1} a_{2} a_{3} a_{4} . . . . . . a_{n}) = c .$
So, we write,
$\frac{a_{1} + a_{2} + a_{3} + a_{4} + . . . . . . + a_{n}}{n} \geq {[(a_{1} a_{2} a_{3} a_{4} . . . . . . a_{n})]}^{1 / n} a_{1} + a_{2} + a_{3} + a_{4} + . . . . . . + a_{n} \geq n {[c]}^{1 / n}$

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If a1,a2,a3,.....,an are positive real numbers whose product is a fixed number c, then the minimum value of a1+a2+a3+.....+an−1+an is

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If $a_{1}, a_{2}, a_{3}, . . . . ., a_{n}$ are positive real numbers whose product is a fixed number $c$ , then the minimum value of $a_{1} + a_{2} + a_{3} + . . . . . + a_{n - 1} + a_{n}$ is