First slide

Arithmetic progression

Question

If three positive real numbers $a, b, c$ are in A.P. such that $a b c = 4$ , then the minimum possible value of $b$ is

Moderate

A

$2^{3 / 2}$
B

$2^{2 / 3}$
C

$2^{1 / 3}$
D

$2^{5 / 2}$

Solution

Let $d$ be the common difference of the A.P., then
$\begin{array}{l} 4 = a b c = (b - d) b (b + d) = b (b^{2} - d^{2}) \\ \Rightarrow b^{3} = 4 + b d^{2} \geq 4 [∵ b > 0, d^{2} \geq 0] \\ \Rightarrow b \geq 2^{2 / 3} \end{array}$
Thus, minimum possible value of $b$ is $2^{2 / 3}$ , that is the case when $d = 0 .$

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