Question

If p^th, q^th and r^th terms of a G.P. are x, y, z respectively, then write the value of x^{q − r} y^{r − p} z^{p − q}.

Answer 1

Let us take a G.P. whose first term is A and common ratio is R.
$$\begin{gathered} \mathrm{According}\mathrm{to}\mathrm{the}\mathrm{question},\mathrm{we}\mathrm{have}: \\ A{R}^{p\mathit{-}\mathit{1}}\mathit{}\mathit{=}\mathit{}x \\ A{R}^{q\mathit{-}\mathit{1}}\mathit{}\mathit{=}\mathit{}y \\ A{R}^{r\mathit{-}\mathit{1}}\mathit{}\mathit{=}\mathit{}z \\ \therefore x^{q\mathit{-}r\mathit{}}{y}^{r\mathit{-}p}\mathit{}{z}^{p\mathit{-}q} \\ \mathit{=}\mathit{}{A}^{q\mathit{-}r}\times \mathit{}{R}^{\left(p\mathit{-}\mathit{1}\right)\left(q\mathit{-}r\right)}\times A^{r-p}\times R^{\left(q\mathit{-}\mathit{1}\right)\left(r\mathit{-}p\right)}\mathit{}\times A^{p-q}\times R^{\left(r-1\right)\left(p-q\right)} \\ =A^{q-r+r-p+p-q}\times R^{\left(pr-pr-q+r\right)+\left(rq-r+p-pq\right)+\left(pr-p-qr+q\right)} \\ =A^0\times R^0 \\ =1 \\ \therefore x^{q\mathit{-}r\mathit{}}{y}^{r\mathit{-}p}\mathit{}{z}^{p\mathit{-}q}=1 \end{gathered}$$