First slide

Harmonic Progression

Question

If $a, b, c$ are in A.P. $p, q, r$ are in H.P. and $a p, b q, c r$ are in G.P., then $\frac{p}{r} + \frac{r}{p}$ is equal to

Moderate

A

$\frac{a}{c} + \frac{c}{a}$
B

$\frac{a}{c} - \frac{c}{a}$
C

$\frac{b}{q} + \frac{q}{b}$
D

$\frac{b}{q} - \frac{q}{p}$

Solution

According to the given condition
$2 b = a + c, q = \frac{2 p r}{p + r}$ and $b^{2} q^{2} = (a p) (c r)$
Substituting values of $b$ and $q$ from first two expressions in the last expression, we get
$\begin{array}{l} \frac{(a + c)^{2}}{4} {(\frac{2 p r}{p + r})}^{2} = (a c) (p r) \\ \Rightarrow \frac{(a + c)^{2}}{a c} = \frac{(p + r)^{2}}{p r} \Rightarrow \frac{a}{c} + \frac{c}{a} = \frac{p}{r} + \frac{r}{p} \end{array}$

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