From the question it is given that,
Common ratio r =
![\[3\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-df8bcc09db12acd0f6bda88b4b8135fb_l3.png "Rendered by QuickLaTeX.com")
![\[{{S}_{7}}~=\text{ }2186\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-a33fda2fd9c4792be1d0fc54a87e5624_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")
![\[\begin{array}{*{35}{l}} {{S}_{7}}~=\text{ }a({{3}^{n}}~\text{ }1)/\left( 3\text{ }\text{ }1 \right) \\ 2186\text{ }=\text{ }a({{3}^{n}}~\text{ }1)/2 \\ \end{array}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-13e78d58092a1a83f2702cc9cf8eb4d8_l3.png "Rendered by QuickLaTeX.com")
By cross multiplication,
![\[\begin{array}{*{35}{l}} \left( 2186\text{ }\times \text{ }2 \right)\text{ }=\text{ }a({{3}^{7}}~\text{ }1) \\ \left( 4372 \right)\text{ }=\text{ }a\left( 2187\text{ }\text{ }1 \right) \\ 4372\text{ }=\text{ }a2186 \\ a\text{ }=\text{ }4372/2186 \\ a\text{ }=\text{ }2 \\ \end{array}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-9cc58c449af63df3028e64f805e0708c_l3.png "Rendered by QuickLaTeX.com")
Then,
![\[{{a}_{3}}~=\text{ }a{{r}^{3\text{ }\text{ }1}}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-8bae6732d2f0f05d74be6b3389349966_l3.png "Rendered by QuickLaTeX.com")
![\[\begin{array}{*{35}{l}}</strong> <strong> =\text{ }a{{r}^{2}} \\</strong> <strong> =\text{ }2\text{ }\times \text{ }{{3}^{2}} \\</strong> <strong> =\text{ }2\text{ }\times \text{ }9 \\</strong> <strong> {{a}_{3}}~=\text{ }18 \\</strong> <strong>\end{array}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-fb1265e1395820aa2b71e29ae31d7d32_l3.png "Rendered by QuickLaTeX.com")
Find the third term of a G.P. whose common ratio is
![\[2186\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-d8180eabb512370ecb26cf5ce287f3d0_l3.png "Rendered by QuickLaTeX.com")
From the question it is given that,
Common ratio r =
We know that,
By cross multiplication,
Then,
