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Home » If the first term of a G.P. is     and the sum of first three terms is     , find the common ratio.
If the first term of a G.P. is

    \[5\]

and the sum of first three terms is

    \[31/5\]

, find the common ratio.
Arithmetic and Geometric Progression | Class 10 | Exercise 1 | ICSE | Maths | ML Aggarwal
If the first term of a G.P. is

    \[5\]

and the sum of first three terms is

    \[31/5\]

, find the common ratio.

From the question it is given that,

First term of a G.P. is a =

    \[5\]

The sum of first three terms is S3 =

    \[31/5\]

We know that,

    \[{{S}_{n}}~=\text{ }a({{r}^{n}}~\text{ }1)/\left( r\text{ }\text{ }1 \right)\]

    \[\begin{array}{*{35}{l}} {{S}_{3}}~=\text{ }a({{r}^{3}}~\text{ }1)/\left( r\text{ }\text{ }1 \right) \\ 31/5\text{ }=\text{ }5({{r}^{3}}~\text{ }1)/\left( r\text{ }\text{ }1 \right) \\ 31/\left( 5\text{ }\times \text{ }5 \right)\text{ }=\text{ }({{r}^{3}}~\text{ }1)/\left( r\text{ }\text{ }1 \right) \\ 31/25\text{ }=\text{ }({{r}^{3}}~\text{ }1)/\left( r\text{ }\text{ }1 \right) \\ \left( r\text{ }\text{ }1 \right)\text{ }({{r}^{2}}~+\text{ }r\text{ }+\text{ }1)/\left( r\text{ }\text{ }1 \right)\text{ }=\text{ }31/25 \\ {{r}^{2}}~+\text{ }r\text{ }+\text{ }1\text{ }=\text{ }31/25 \\ \end{array}\]

By cross multiplication we get,

    \[\begin{array}{*{35}{l}} 25({{r}^{2}}~+\text{ }r\text{ }+\text{ }1)\text{ }=\text{ }31 \\ 25{{r}^{2}}~+\text{ }25r\text{ }+\text{ }25\text{ }=\text{ }31 \\ \end{array}\]

Transposing 31 from right hand side to left hand side it becomes

    \[-31\]

,

    \[\begin{array}{*{35}{l}} 25{{r}^{2}}~+\text{ }25r\text{ }+\text{ }25\text{ }\text{ }31\text{ }= \\ 25{{r}^{2}}~+\text{ }25r\text{ }\text{ }6\text{ }=\text{ }0 \\ 25{{r}^{2}}~+\text{ }30r\text{ }\text{ }5r\text{ }\text{ }6\text{ }=\text{ }0 \\ 5r\left( 5r\text{ }+\text{ }6 \right)\text{ }\text{ }1\left( 5r\text{ }+\text{ }6 \right)\text{ }=\text{ }0 \\ \left( 5r\text{ }\text{ }1 \right)\text{ }\left( 5r\text{ }+\text{ }6 \right)\text{ }=\text{ }0 \\ \end{array}\]

Take

    \[5r\text{ }\text{ }1\text{ }=\text{ }0\]

    \[\begin{array}{*{35}{l}} r\text{ }=\text{ }1/5 \\ or\text{ }5r\text{ }+\text{ }6\text{ }=\text{ }0 \\ r\text{ }=\text{ }-6/5 \\ \end{array}\]

Therefore, common ratio

    \[r\text{ }=\text{ }1/5\text{ }or\text{ }-6/5\]

.

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