If the $p^{\text {th }}, q^{\text {th }}$ and $r^{\text {th }}$ terms of a $G.P.$ are $a, b$ and $c,$ respectively. Prove that
$a^{q-r} b^{r-p} c^{p-q}=1$
If the $p^{\text {th }}, q^{\text {th }}$ and $r^{\text {th }}$ terms of a $G.P.$ are $a, b$ and $c,$ respectively. Prove that
$a^{q-r} b^{r-p} c^{p-q}=1$
Let $A$ be the first term and $R$ be the common ratio of the $G.P.$
According to the given information,
$A R^{p-1}=a$
$A R^{q-1}=b$
$A R^{r-1}=c$
$a^{q-r} \cdot b^{r-p} \cdot c^{p-q}$
$=A^{q-r} \times R^{(p-1)(q-r)} \times A^{r-p} \times R^{(q-1)(r-p)} \times A^{p-q} \times R^{(r-1)(p-q)}$
$ = {A^{q - r + r - p + p - q}} \times {R^{(pr - pr - q + r) + (rq - r + p - pq) + (pr - p - qr + q)}}$
$=A^{0} \times R^{0}$
$=1$
Thus, the given result is proved.
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