First slide

Harmonic Progression

Question

$If \frac{1}{a} + \frac{1}{a - 2 b} + \frac{1}{c} + \frac{1}{c - 2 b} = 0 and a, b, c are not in A.P then:$

Moderate

A

$a, b, c are in G.P$
B

$a, \frac{b}{2}, c are in A.P$
C

$a, \frac{b}{2}, c are in H.P$
D

$a, 2 b, c are in H.P$

Solution

$\begin{array}{l} \frac{1}{a} + \frac{1}{a - 2 b} + \frac{1}{c} + \frac{1}{c - 2 b} = 0 \\ \frac{2 (a - b)}{a (a - 2 b)} + \frac{2 (c - b)}{c (c - 2 b)} = 0 \\ \frac{a - b}{a (a - 2 b)} + \frac{c - b}{c (c - 2 b)} = 0 \\ Now, we can verify the result that a, 2 b, c are in H.P \end{array}$

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