The positive numbers a 1 , a 2 , a 3 are in A.P. such that a 1 a 2 a 3 = 64 . Then the minimum value of a 2 is

Question

The positive numbers $a_{1}, a_{2}, a_{3}$ are in A.P. such that $a_{1} a_{2} a_{3} = 64$ . Then the minimum value of $a_{2}$ is

Smart Achievers · Accepted Answer

(1)

$From A.M.-G.M. theorem$

$\frac{a_{1} + a_{2} + a_{3}}{3} \geq \sqrt[3]{a_{1} a_{2} a_{3}}$

$\Rightarrow$ $a_{1} + a_{2} + a_{3} \geq 3 \sqrt[3]{64}$

$\Rightarrow$ $a_{1} + a_{2} + a_{3} \geq 12$ ...(i)

$Now, a_{2} is the mean of a_{1} and a_{3}$

$∴$ $a_{1} + a_{3} = 2 a_{2}$ ...(ii)

$So from (i) and (ii)$

$2 a_{2} + a_{2} \geq 12 \Rightarrow 3 a_{2} \geq 12 \Rightarrow a_{2} \geq 4$

$∴$ ${(a_{2})}_{\min} = 4$

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