Question

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

Answer 1

Let $a$ be the first term and $r$ be the common ratio of the G.P.

Thus according to the given information,

${ar}^{p-1}=a,{ar}^{p-1}=b,{ar}^{p-1}=c$

$\therefore a^{q-r}{b}^{r-p}{c}^{p-q}$

$=a^{q-r}\times r^{(p-1)(q-r)}\times a^{r-p}\times r^{(q-1)(r-p)}\times a^{p-q}\times r^{(r-1)(p-q)}$

$=a^{-r+r-p+p-q}\times r^{(pr-pr-q+r)+(rq-r+p-pq)+(pr-p-qr+q)}$

$=a^0\cdot r^0=1$. $\left[henceproved\right]$