If the $p^{t h} . q^{t h}$ and $r^{t h}$ terms of a $G . P$ are $a, b,$ and $c$ respectively. prove that $a^{q - r} b^{r - p} c^{p - q} = 1$ .

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Solution

Let first term of $G P = K$
common ratio $= m$
$p t h$ term $= K m^{p - 1} = a$
$q t h$ term $= K m^{q - 1} = b$
$r t h$ term $K m^{r - 1} = c$
$a^{q - r} . b^{r - p} . c^{p - q} = \frac{a^{q}}{a^{r}} . \frac{b^{r}}{b^{p}} . \frac{c^{p}}{c^{q}} = \frac{K^{q} {(m^{p - 1})}^{q}}{K^{r} {(m^{p - 1})}^{r}} . \frac{K^{r} . {(m^{q - 1})}^{r}}{K^{p} {(m^{q - 1})}^{p}} . \frac{K^{p} {(m^{r - 1})}^{p}}{K^{q} {(m^{r - 1})}^{q}} = \frac{m^{p q} . m^{r}}{m^{q} . m^{p r}} . \frac{m^{q r} . m^{p}}{m^{p q} . m^{r}} . \frac{m^{p r} . m^{q}}{m^{q r} . m^{p}} = 1$

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