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Question
Find an infinite G.P. whose first term is 1 and each term is the sum of all the terms which follow it.
Solution
Here, first term, a = 1
Common ratio = r
\[\therefore a_n = \left[ a_{n + 1} + a_{n + 2} + a_{n + 3} + . . . . . \infty \right] \forall n \in N\]
\[ \Rightarrow a r^{n - 1} = a r^n + a r^{n - 1} + . . . . . \infty \]
\[ \Rightarrow r^{n - 1} = \frac{r^n}{1 - r} \left[ \text { Putting a } = 1 \right]\]
\[ \Rightarrow r^{n - 1} \left( 1 - r \right) = r^n \]
\[ \Rightarrow 1 - r = r\]
\[ \Rightarrow 2r = 1 \]
\[ \Rightarrow r = \frac{1}{2}\]
\[\text { Thus, the infinte G . P is } 1, \frac{1}{2}, \frac{1}{4}, . . . \infty .\]
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