Means - A.M/G.M/H.M

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If a,b,c are positive numbers in A.P such that their product is 64 then the minimum value of ‘b’ is

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Solution

$\begin{array}{l} abc=64 \\ 2 b = a + c (since a, b, c are in A . P) \\ \Rightarrow \frac{a + b + c}{3} \geq \sqrt[3]{abc} (\sin c e A . M \geq G . M) \\ \Rightarrow \frac{3 b}{3} \geq \sqrt[3]{64} \Rightarrow b \geq 4 \end{array}$

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