First slide

Harmonic Progression

Question

$Suppose 'a' is a fixed real number such that \frac{a - x}{p x} = \frac{a - y}{q y} = \frac{a - z}{r z}$ $if p, q, r are in AP then x, y, z all are in$

Moderate

A

A.P
B

G.P
C

H.P
D

A.G.P

Solution

$p - q = q - r = k (l e t) \Rightarrow \frac{\frac{a}{x} - 1}{p} = \frac{\frac{a}{y} - 1}{q} = \frac{\frac{a}{z} - 1}{r}$
$\begin{aligned} \Rightarrow \frac{(\frac{a}{x} - 1) - (\frac{a}{y} - 1)}{p - q} = \frac{(\frac{a}{y} - 1) - (\frac{a}{z} - 1)}{q - r} \\ By law of proportion \Rightarrow \frac{1}{x} - \frac{1}{y} = \frac{1}{y} - \frac{1}{z} \end{aligned}$

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