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Standard XII

Mathematics

Variable Separable Method

If a1+a2+a3...

Question

If $a_{1} + a_{2} + a_{3} + a_{4} + a_{5} + . . . . . . . a_{n} = 1$ for all $a_{i} > 0, i = 1, 2, 3....... n .$ Then the maximum value of $a_{1} a_{2} a_{3} a_{4} a_{5} . . . . . . . . . . a_{n}$ is

\frac{2}{(n + 1)^{n}}

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\frac{2}{(n + 1)^{n + 1}}

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\frac{2}{n^{n}}

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\frac{4}{n^{n + 1}}

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Solution

The correct option is B $\frac{2}{(n + 1)^{n + 1}}$
$\frac{a_{1} + a_{2} + a_{3} + . . . . . + a_{n} + 1}{n + 1} \geq (a_{1} a_{2} a_{3} . . . . . a_{n} .1)^{1 / n}$ (Since, $A M \geq G M$ for positive quantities.)
$\frac{1 + 1}{n + 1} \geq (a_{1} a_{2} a_{3} . . . . . a_{n} .1)^{1 / n}$
$(\frac{2}{(n + 1)^{n}}) \geq (a_{1} a_{2} a_{3} . . . . . a_{n})$

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