We should take
![\[A\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-362a7fe0462e59a454faefa8c7031586_l3.png "Rendered by QuickLaTeX.com")
to be the initial term and
![\[R\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-856a54b7cde51a14556b6d5d43eee08a_l3.png "Rendered by QuickLaTeX.com")
to be the normal proportion of the G.P.
Then, at that point, as per the inquiry, we have
![\[\begin{array}{*{35}{l}} A{{R}^{p}}^{1~}=~a \\ A{{R}^{q}}^{1~}=~b \\ A{{R}^{r}}^{1~}=~c \\ Then, \\ {{a}^{qr}}^{~}{{b}^{rp}}^{~}{{c}^{pq}} \\ =~{{A}^{q}}{{^{}}^{r~}}\times ~{{R}^{(}}{{^{p}}^{1)\text{ }(qr)}}~\times \text{ }{{A}^{r}}{{^{}}^{p}}~\times ~{{R}^{(}}{{^{q}}^{1)\text{ }(}}{{^{r}}^{}}{{^{p}}^{)}}~\times ~{{A}^{p}}{{^{}}^{q}}~\times ~{{R}^{(}}{{^{r~}}^{1)(}}{{^{p}}^{}}{{^{q}}^{)}} \\ =~A{{q}^{~~}}{{^{r}}^{~+~}}{{^{r}}^{~~}}{{^{p}}^{~+~}}{{^{p}}^{~~}}^{q}~\times ~R{{~}^{(}}{{^{pr}}^{~~}}{{^{pr}}^{~~}}{{^{q}}^{~+~}}{{^{r}}^{)\text{ }+\text{ }(}}{{^{rq}}^{~}}{{^{~r~}}^{+~}}{{^{p}}^{~~}}{{^{pq}}^{)\text{ }+\text{ }(}}{{^{pr}}^{~~}}{{^{p}}^{~~}}{{^{qr}}^{~+~}}{{^{q}}^{)}} \\ =~{{A}^{0}}~\times ~{{R}^{0}} \\ =\text{ }1 \\ Hence\text{ }proved. \\ \end{array}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-6a6e861c1be9dd9ea4c7da83639e876e_l3.png "Rendered by QuickLaTeX.com")
If the pth, qth and rth terms of a G.P. are a, b and c, respectively. Prove that aq-r br-p cp-q = 1
If the pth, qth and rth terms of a G.P. are a, b and c, respectively. Prove that aq-r br-p cp-q = 1