First slide

Arithmetic progression

Question

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

Moderate

A

$2^{2 / 3}$
B

$2^{1 / 3}$
C

$2^{5 / 3}$
D

None of these

Solution

Let d be the common difference of the A.P., then 4 = abc = (b – d)b (b + d) = b(b² – d² )
$\begin{array}{ll} \Rightarrow & b^{3} = 4 + {bd}^{2} \geq 4 (∵ b > 0, d^{2} \geq 0) \\ \Rightarrow & b \geq 2^{2 / 3} \end{array}$
Thus, the minimum possible value of b is 2^2/3.

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