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If { a }_{ 1 },{ a }_{ 2 },....{ a }_{ n } are positive real n

Question

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

n (2 c)^{1/ n}

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

2 n c^{1/ n}

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

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Henry Discussed

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}

8 mins ago

Discuss this question LIVE

8 mins ago

Text solutionVerified

From

A . M \geq G . M

we have

\Rightarrow \frac{a _{1} + a _{2} + .... + a _{n - 1} + 2 a _{n}}{n} \geq (a_{1} a_{2} .... a_{n - 1} 2 a_{n})^{1/ n}

\Rightarrow \frac{a _{1} + a _{2} + .... + a _{n - 1} + 2 a _{n}}{n} \geq (2 c)^{1/ n}

\Rightarrow a_{1} + a_{2} + .... + a_{n - 1} + 2 a_{n} \geq n (2 c)^{1/ n}

The minimum value is

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

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Question Text	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
Answer Type	Text solution:1
Upvotes	150