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Q.
If $\frac{a-x}{p x}=\frac{a-y}{q y}=\frac{a-z}{r z}$ and $p, q, r$ be in A.P., then $x, y$, $z$ are in
Sequences and Series
Solution:
We have, $\frac{a-x}{p x}=\frac{a-y}{q y}=\frac{a-z}{r z}=\lambda$ (say)
$\Rightarrow p=\frac{a-x}{\lambda x}, q=\frac{a-y}{\lambda y}, r=\frac{a-z}{\lambda z}$
Now $p, q, r$ are in $A.P$.
$\Rightarrow \frac{a-x}{\lambda x}, \frac{a-y}{\lambda y}, \frac{a-z}{\lambda z}$ are in $A.P$.
$\Rightarrow \frac{a-x}{x}, \frac{a-y}{y}, \frac{a-z}{z}$ are in $A.P$.
$\Rightarrow \frac{a}{x}-1, \frac{a}{y}-1, \frac{a}{z}-1$ are in $A.P$.
$\Rightarrow \frac{a}{x}, \frac{a}{y}, \frac{a}{z}$ are in $A.P.$
$\Rightarrow \frac{1}{x}, \frac{1}{y}, \frac{1}{z}$ are in $A.P.$
$\Rightarrow x, y, z$ are in $HP.$