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Question
If the sum of first two terms of an infinite GP is 1 every term is twice the sum of all the successive terms, then its first term is
Options
(a) 1/3
(b) 2/3
(c) 1/4
(d) 3/4
Solution
(d) 3/4
\[\text{ Let the terms of the G . P } . be a, a_2 , a_3 , a_4 , a_5 , . . . , \infty . \]
\[\text{ And, let the common ratio be r } . \]
\[\text{ Now }, a + a_2 = 1\]
\[ \therefore a + ar = 1 . . . . . . . . (i)\]
\[\text{ Also }, a = 2\left( a_2 + a_3 + a_4 + a_5 + . . . \infty \right)\]
\[ \Rightarrow a = 2\left( ar + a r^2 + a r^3 + a r^4 + . . . \infty \right)\]
\[ \Rightarrow a = 2\left( \frac{ar}{1 - r} \right)\]
\[ \Rightarrow 1 - r = 2r\]
\[ \Rightarrow 3r = 1\]
\[ \Rightarrow r = \frac{1}{3}\]
\[\text{ Putting the value of r in } (i): \]
\[a + \frac{a}{3} = 1\]
\[ \Rightarrow \frac{4a}{3} = 1\]
\[ \Rightarrow 4a = 3\]
\[ \Rightarrow a = \frac{3}{4}\]
\[\]
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