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.

Answer 1

Let the first term be a and the common difference be r.

\[\therefore a_1 + a_2 = 5 \]

\[ \Rightarrow a + ar = 5 . . . \left( i \right)\]

\[\text { Also, } a_n = 3\left[ a_{n + 1} + a_{n + 2} + a_{n + 3} + . . . \infty \right] \forall n \in N\]

\[ \Rightarrow a r^{n - 1} = 3 \left[ a r^{n + 1} + a r^{n + 2} + a r^{n + 3} + . . . \infty \right]\]

\[ \Rightarrow a r^{n - 1} = \frac{3a r^n}{1 - r} \]

\[ \Rightarrow 1 - r = 3r\]

\[ \Rightarrow 4r = 1 \]

\[ \Rightarrow r = \frac{1}{4}\]

\[\text { Putting } r = \frac{1}{4} \text { in } \left( i \right): \]

\[a + \frac{a}{4} = 5\]

\[ \Rightarrow 5a = 20 \]

\[ \Rightarrow a = 4\]

\[\text { Thus, the G . P . is } 4, 1, \frac{1}{4}, \frac{1}{16}, . . . \infty . \]