Multiple choice

The first term of an infinite G.P. is $1$ and any term is equal to the sum of all the succeeding terms. Find the series.

  1. $1, 2, 4, 8, ......$
  2. $1,\, \displaystyle {\frac{1}{3},\, \frac{1}{9},\, \frac{1}{27},\, .....}$
  3. $1,\, \displaystyle {\frac{1}{4},\, \frac{1}{8}}, .....$
  4. $1,\, \displaystyle {\frac{1}{2},\, \frac{1}{4},\, \frac{1}{8},\, .....}$
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D Correct answer
Explanation

For a G.P. with first term a=1 and common ratio r, the sum of succeeding terms is a*r / (1-r). Setting 1 = r / (1-r) gives 1-r = r, so 2r = 1, r = 1/2. The series is 1, 1/2, 1/4, 1/8, ...

AI explanation

In a geometric progression, the sum of all succeeding terms after any term T is Tr/(1 - r). If T equals this sum, then T = Tr/(1 - r), which simplifies to 1 = r/(1 - r) and yields a common ratio r of 1/2. The series must start with 1 and have a common ratio of 1/2, making the sequence 1, 1/2, 1/4, 1/8, and so on.

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