First slide

Arithmetic progression

Question

$\frac{1}{q + r}, \frac{1}{r + p}, \frac{1}{p + q}$ are in A.P., then:

Easy

A

$p, q, r$ are in A.P.
B

$p^{2}, q^{2}, r^{2}$ are in A.P.
C

$1 / p, 1 / q, 1 / r$ are in A.P.
D

None of these

Solution

$1 / (q + r), 1 / (r + p), 1 / (p + q)$ are in A.P.
$\Rightarrow$ $\frac{1}{r + p} - \frac{1}{q + r} = \frac{1}{p + q} - \frac{1}{r + p}$
$\Rightarrow \frac{q - p}{(r + p) (r + q)} = \frac{r - q}{(p + q) (r + p)}$
$\Rightarrow$ $q^{2} - p^{2} = r^{2} - q^{2}$ $\Rightarrow p^{2}, q^{2}, r^{2}$ are in A.P.

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