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Question
The sum of first two terms of an infinite G.P. is 5 and each term is three times the sum of the succeeding terms. Find the G.P.
Solution
Let the first term be a and the common difference be r.
\[\therefore a_1 + a_2 = 5 \]
\[ \Rightarrow a + ar = 5 . . . \left( i \right)\]
\[\text { Also, } a_n = 3\left[ a_{n + 1} + a_{n + 2} + a_{n + 3} + . . . \infty \right] \forall n \in N\]
\[ \Rightarrow a r^{n - 1} = 3 \left[ a r^{n + 1} + a r^{n + 2} + a r^{n + 3} + . . . \infty \right]\]
\[ \Rightarrow a r^{n - 1} = \frac{3a r^n}{1 - r} \]
\[ \Rightarrow 1 - r = 3r\]
\[ \Rightarrow 4r = 1 \]
\[ \Rightarrow r = \frac{1}{4}\]
\[\text { Putting } r = \frac{1}{4} \text { in } \left( i \right): \]
\[a + \frac{a}{4} = 5\]
\[ \Rightarrow 5a = 20 \]
\[ \Rightarrow a = 4\]
\[\text { Thus, the G . P . is } 4, 1, \frac{1}{4}, \frac{1}{16}, . . . \infty . \]
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