If pth, qth and rth term of a GP are a, b, c respectively, then $a^{q-r}\cdot b^{r-p}\cdot c^{p-q}$ is equal to 

Given, pth term= a, qth term= band zth term= c 

Let A be the first term, R be the common difference of GP, then

$\begin{array}{r}a=A{R}^{p-1}\\ b=A{R}^{q-1}\\ c=A{R}^{\mathrm{\prime }-1}\end{array}$

Now, $a^{q-r}\cdot b^{r-\rho }\cdot C^{p-q}={\left(A{R}^{\rho -1}\right)}^{q-r}{\left(A{R}^{q-1}\right)}^{r-p}$

${\left(A{R}^{r-1}\right)}^{\rho -q}$ 

$\begin{array}{l}=A^{q-r+r-\rho +p-q}\\ R^{(\rho -1)(q-r)+(q-1)(r-p)+(r-1)(\rho -q)}\\ =A^0{R}^{pq-pr-q+r+qr-pq-r+\rho +rp-rq-p+q}\\ =A^0{R}^0=(AR{)}^0=1\end{array}$