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Question
Show that one of the following progression is a G.P. Also, find the common ratio in case:1/2, 1/3, 2/9, 4/27, ...
Solution
We have,
\[ a_1 = \frac{1}{2} , a_2 = \frac{1}{3}, a_3 = \frac{2}{9}, a_4 = \frac{4}{27}\]
\[\text { Now }, \frac{a_2}{a_1} = \frac{\frac{1}{3}}{\frac{1}{2}} = \frac{2}{3}, \frac{a_3}{a_2} = \frac{\frac{2}{9}}{\frac{1}{3}} = \frac{2}{3}, \frac{a_4}{a_3} = \frac{\frac{4}{27}}{\frac{2}{9}} = \frac{2}{3}\]
\[ \therefore \frac{a_2}{a_1} = \frac{a_3}{a_2} = \frac{a_4}{a_3} = \frac{2}{3}\]
\[\text { Thus, } a_1 , a_2 , a_3 \text { and } a_4 \text { are in G . P . , where the first term is} \frac{1}{2} \text { and the common ratio is } \frac{2}{3} .\]
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