From the question it is given that,
First term of a G.P. is a =
![\[5\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-7dc8465919d0838925dbc6c99670258c_l3.png "Rendered by QuickLaTeX.com")
The sum of first three terms is S3 =
![\[31/5\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-9e999b159ba597f5bdf395a7852b6c8d_l3.png "Rendered by QuickLaTeX.com")
We know that,
![\[{{S}_{n}}~=\text{ }a({{r}^{n}}~\text{ }1)/\left( r\text{ }\text{ }1 \right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-51a336950c04b131c785a9df0ddee440_l3.png "Rendered by QuickLaTeX.com")
![\[\begin{array}{*{35}{l}} {{S}_{3}}~=\text{ }a({{r}^{3}}~\text{ }1)/\left( r\text{ }\text{ }1 \right) \\ 31/5\text{ }=\text{ }5({{r}^{3}}~\text{ }1)/\left( r\text{ }\text{ }1 \right) \\ 31/\left( 5\text{ }\times \text{ }5 \right)\text{ }=\text{ }({{r}^{3}}~\text{ }1)/\left( r\text{ }\text{ }1 \right) \\ 31/25\text{ }=\text{ }({{r}^{3}}~\text{ }1)/\left( r\text{ }\text{ }1 \right) \\ \left( r\text{ }\text{ }1 \right)\text{ }({{r}^{2}}~+\text{ }r\text{ }+\text{ }1)/\left( r\text{ }\text{ }1 \right)\text{ }=\text{ }31/25 \\ {{r}^{2}}~+\text{ }r\text{ }+\text{ }1\text{ }=\text{ }31/25 \\ \end{array}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-724d58d0cc806e937eccf5a41bd009a7_l3.png "Rendered by QuickLaTeX.com")
By cross multiplication we get,
![\[\begin{array}{*{35}{l}} 25({{r}^{2}}~+\text{ }r\text{ }+\text{ }1)\text{ }=\text{ }31 \\ 25{{r}^{2}}~+\text{ }25r\text{ }+\text{ }25\text{ }=\text{ }31 \\ \end{array}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-747ad3851e9f7bcd96b1294fab1f55da_l3.png "Rendered by QuickLaTeX.com")
Transposing 31 from right hand side to left hand side it becomes
![\[-31\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-db6a50add5cd648794f86de2c1e0ec1c_l3.png "Rendered by QuickLaTeX.com")
,
![\[\begin{array}{*{35}{l}} 25{{r}^{2}}~+\text{ }25r\text{ }+\text{ }25\text{ }\text{ }31\text{ }= \\ 25{{r}^{2}}~+\text{ }25r\text{ }\text{ }6\text{ }=\text{ }0 \\ 25{{r}^{2}}~+\text{ }30r\text{ }\text{ }5r\text{ }\text{ }6\text{ }=\text{ }0 \\ 5r\left( 5r\text{ }+\text{ }6 \right)\text{ }\text{ }1\left( 5r\text{ }+\text{ }6 \right)\text{ }=\text{ }0 \\ \left( 5r\text{ }\text{ }1 \right)\text{ }\left( 5r\text{ }+\text{ }6 \right)\text{ }=\text{ }0 \\ \end{array}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-91d2184c7e1bb43e9d81b2cbcfee4fc2_l3.png "Rendered by QuickLaTeX.com")
Take
![\[5r\text{ }\text{ }1\text{ }=\text{ }0\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-5cd4bed013f1509be67aeaf53396a752_l3.png "Rendered by QuickLaTeX.com")
![\[\begin{array}{*{35}{l}} r\text{ }=\text{ }1/5 \\ or\text{ }5r\text{ }+\text{ }6\text{ }=\text{ }0 \\ r\text{ }=\text{ }-6/5 \\ \end{array}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-21d871463c5d1fd6ac8ec1e86b0e34ee_l3.png "Rendered by QuickLaTeX.com")
Therefore, common ratio
![\[r\text{ }=\text{ }1/5\text{ }or\text{ }-6/5\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-1834d415f57a8fb3c2f439a70a829997_l3.png "Rendered by QuickLaTeX.com")
.