First slide

Harmonic Progression

Question

Suppose $a, b, c$ are positive real numbers. If $a, b, c$ are in A.P. and $a^{2}, b^{2}, c^{2}$ are in H.P., then

Moderate

A

$a = b = c$
B

$2 b = 3 a + c$
C

$b^{2} = \sqrt{a c / 8}$
D

none of these

Solution

We have $2 b = a + c$ and $b^{2} = \frac{2 a^{2} c^{2}}{a^{2} + c^{2}}$
$\begin{array}{l} ∴ {(\frac{a + c}{2})}^{2} = \frac{2 a^{2} c^{2}}{a^{2} + c^{2}} \Rightarrow (a + c)^{2} (a^{2} + c^{2}) = 8 a^{2} c^{2} \\ \Rightarrow {(a^{2} + c^{2})}^{2} + 2 a c (a^{2} + c^{2}) - 8 a^{2} c^{2} = 0 \end{array}$
$\begin{array}{ll} \Rightarrow (a^{2} + c^{2} + 4 a c) (a^{2} + c^{2} - 2 a c) = 0 \\ \Rightarrow [(a + c)^{2} + 2 a c] (a - c)^{2} = 0 \end{array}$
$\Rightarrow a - c = 0$ or $(a + c)^{2} + 2 a c = 0$
As ${(a + c)}^{2} + 2 a c > 0,$ we get $a = c \Rightarrow a = b = c$

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