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Inequality

If a1, a2, ...

Question

If $a_{1}, a_{2}, a_{3} . . . . a_{n} \in R^{+}$ and $a_{1} . a_{2} . a_{3} . . . . a_{n} = 1$ , then minimum value of $(1 + a_{1} + a_{1}^{2}) (1 + a_{2} + a_{2}^{2}) (1 + a_{3} + a_{3}^{2}) . . . . (1 + a_{n} + a_{n}^{2})$ is equal to.

A

3^{n + 1}

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B

3^{n}

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C

3^{n - 1}

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D

none of these

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Solution

The correct option is B $3^{n}$
The given expression will be minimum, when $a_{1} = a_{2} = a_{3} = . . . a_{n}$ .
Now, it is given that
$a_{1} . a_{2} . a_{3} . . . a_{n} = 1$
Hence, $a_{i}^{n} = 1$
Since $a_{i}$ is Real, therefore $a_{i} = 1$
Hence, $(1 + a_{1} + a_{1}^{2}) (1 + a_{2} + a_{2}^{2}) . . . (1 + a_{n} + a_{n}^{2})$
$= (1 + a_{i} + a_{i}^{2})^{n}$
$= 3^{n}$ .

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Similar questions

a_{r} > 0, r \in N

a_{1}, a_{2}, a_{3}, . . ., a_{2 n}

are in A.P. then

\frac{a_{1} + a_{2 n}}{\sqrt{a_{1}} + \sqrt{a_{2}}} + \frac{a_{2} + a_{2 n - 1}}{\sqrt{a_{2}} + \sqrt{a_{3}}} + \frac{a_{3} + a_{2 n - 2}}{\sqrt{a_{3}} + \sqrt{a_{4}}} + . . . + \frac{a_{n} + a_{n + 1}}{\sqrt{a_{n}} + \sqrt{a_{n + 1}}}

a_{r} > 0, r ϵ N

a_{1}

a_{2}

a_{3}

a_{2 n}

\frac{a_{1} + a_{2 n}}{\sqrt{a_{1}} + \sqrt{a_{2}}} + \frac{a_{2} + a_{2 n - 1}}{\sqrt{a_{2}} + \sqrt{a_{3}}} + \frac{a_{3} + a_{2 n - 2}}{\sqrt{a_{3}} + \sqrt{a_{4}}} + . . . + \frac{a_{n} + a_{n + 1}}{\sqrt{a_{n}} + \sqrt{a_{n + 1}}}

a_{r} > 0; \forall r, n \in N

a_{1}, a_{2}, a_{3}, . . . . . a_{2 n}

are in A.P, then

\frac{a_{1} + a_{2 n}}{\sqrt{a_{1}} + \sqrt{a_{2}}} + \frac{a_{2} + a_{2 n - 1}}{\sqrt{a_{2}} + \sqrt{a_{3}}} + \frac{a_{3} + a_{2 n - 2}}{\sqrt{a_{3}} + \sqrt{a_{4}}} + . . . + \frac{a_{n} + a_{n + 1}}{\sqrt{a_{n}} + \sqrt{a_{n + 1}}} =

a_{1}, a_{2}, a_{3} . . . . a_{15} \in R^{+}

a_{1}, a_{2}, a_{3} . . . a_{n} = 1

, then minimum value of

(1 + a_{1} + a_{1}^{2}) (1 + a_{3} + a_{3}^{2}) . . . . (1 + a_{a} + a_{a}^{2})

a_{1}, a_{2}, a_{3}, . . . . a_{2 n + 1}

are in A.P. then

\frac{a_{2 n + 1} - a_{1}}{a_{2 n + 1} + a_{1}} + \frac{a_{2 n} - a_{2}}{a_{2 n} + a_{2}} + . . . + \frac{a_{n + 2} - a_{n}}{a_{n + 2} + a_{n}}

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