From the question it is given that,
The fourth term of a G.P. is the square of its second term
![\[=\text{ }{{a}_{4}}~=\text{}{{({{a}_{2}})}^{2}}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-0df46fb7de6c2ff2871abe2aa56abec7_l3.png "Rendered by QuickLaTeX.com")
The first term
![\[{{a}_{1}}~=\text{ }\text{ }3\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-c22a010629eee2ee545eb3ebeff46e1f_l3.png "Rendered by QuickLaTeX.com")
We know that,
![\[{{a}_{n}}~=\text{ }a{{r}^{n\text{ }\text{ }1}}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-5c4a26b205addff26274971977b92900_l3.png "Rendered by QuickLaTeX.com")
![\[\begin{array}{*{35}{l}} {{a}_{4}}~=\text{ }a{{r}{4\text{ }\text{ }1}} \\ {{a}_{4}}~=\text{ }a{{r}^{3}} \\ {{a}_{2}}~=\text{ }ar \\ \end{array}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-4f5d35359cc3cec341411365dbdb5b52_l3.png "Rendered by QuickLaTeX.com")
Now,
![\[a{{r}^{3}}~=\text{ }{{\left( ar \right)}^{2}}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-886641fa5077ff5eaa99ec7d6d43aeeb_l3.png "Rendered by QuickLaTeX.com")
![\[\begin{array}{*{35}{l}} a{{r}^{3}}~=\text{ }{{a}^{2}}{{r}^{2}} \\ {{r}^{3}}/{{r}^{2}}~=\text{ }{{a}^{2}}/a \\ {{r}^{3\text{ }\text{ }2}}~=\text{ }{{a}^{2\text{ }\text{ }1}}~\ldots \text{ }[from\text{}{{a}^{m}}/{{a}^{n}}~=\text{ }{{a}^{m\text{ }\text{ }n}}] \\ r\text{ }=\text{ }a \\ {{a}_{1~}}=\text{ }-3 \\ {{a}_{7}}~=\text{ }a{{r}^{7\text{ }\text{ }1}} \\ {{a}_{7}}~=\text{ }a{{r}^{6}} \\ =\text{ }-3\text{ }\times \text{ }{{\left( -3 \right)}^{6}} \\ =\text{ }-3\text{ }\times \text{ }729 \\ =\text{ }-2187 \\ \end{array}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-a4c2ac6c01a9f994f6b83a4d69fcd1d1_l3.png "Rendered by QuickLaTeX.com")
Therefore, the
![\[{{7}^{th}}~term\text{ }{{a}_{7}}~=\text{ }-2187\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-e54f59ddd01dec387cca2eb1d769bee2_l3.png "Rendered by QuickLaTeX.com")
The fourth term of a G.P. is the square of its second term and the first term is
![\[-3\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-4adc8c45b73d97c997db1b8d065f32a8_l3.png "Rendered by QuickLaTeX.com")
