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Home » The fourth term of a G.P. is the square of its second term and the first term is     . Find its 7th term.
The fourth term of a G.P. is the square of its second term and the first term is

    \[-3\]

. Find its 7th term.
Arithmetic and Geometric Progression | Class 10 | Exercise 1 | ICSE | Maths | ML Aggarwal
The fourth term of a G.P. is the square of its second term and the first term is

    \[-3\]

. Find its 7th term.

From the question it is given that,

The fourth term of a G.P. is the square of its second term

    \[=\text{ }{{a}_{4}}~=\text{}{{({{a}_{2}})}^{2}}\]

The first term

    \[{{a}_{1}}~=\text{ }\text{ }3\]

We know that,

    \[{{a}_{n}}~=\text{ }a{{r}^{n\text{ }\text{ }1}}\]

    \[\begin{array}{*{35}{l}} {{a}_{4}}~=\text{ }a{{r}{4\text{ }\text{ }1}} \\ {{a}_{4}}~=\text{ }a{{r}^{3}} \\ {{a}_{2}}~=\text{ }ar \\ \end{array}\]

Now,

    \[a{{r}^{3}}~=\text{ }{{\left( ar \right)}^{2}}\]

    \[\begin{array}{*{35}{l}} a{{r}^{3}}~=\text{ }{{a}^{2}}{{r}^{2}} \\ {{r}^{3}}/{{r}^{2}}~=\text{ }{{a}^{2}}/a \\ {{r}^{3\text{ }\text{ }2}}~=\text{ }{{a}^{2\text{ }\text{ }1}}~\ldots \text{ }[from\text{}{{a}^{m}}/{{a}^{n}}~=\text{ }{{a}^{m\text{ }\text{ }n}}] \\ r\text{ }=\text{ }a \\ {{a}_{1~}}=\text{ }-3 \\ {{a}_{7}}~=\text{ }a{{r}^{7\text{ }\text{ }1}} \\ {{a}_{7}}~=\text{ }a{{r}^{6}} \\ =\text{ }-3\text{ }\times \text{ }{{\left( -3 \right)}^{6}} \\ =\text{ }-3\text{ }\times \text{ }729 \\ =\text{ }-2187 \\ \end{array}\]

Therefore, the

    \[{{7}^{th}}~term\text{ }{{a}_{7}}~=\text{ }-2187\]

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