Each term of an infinite geometric progression is twice the...

Each term of an infinite geometric progression is twice the sum of all the terms which follows it. The common ratio of this GP is
(a) $\frac{1}{3}$
(b) $\frac{1}{2}$
(c) $\frac{1}{4}$
(d) $\frac{2}{3}$

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Key Concepts

Convergence Condition

The convergence condition for an infinite geometric series requires that the absolute value of the common ratio is less than one (|r| < 1). This condition ensures that successive terms decrease in magnitude sufficiently fast so that the total sum approaches a finite limit.

Infinite Geometric Series

An infinite geometric series is the sum of the terms of a geometric progression extending indefinitely. When the absolute value of the common ratio is less than one, the series converges to a finite sum, which is calculated using the formula S = a/(1-r), where 'a' is the first term and 'r' is the common ratio.

Geometric Progression

A geometric progression is a sequence of numbers where each term is obtained by multiplying the previous term by a fixed, non-zero number called the common ratio. This property leads to each term being a predictable multiple of its predecessor, making it a foundational concept in sequences and series in mathematics.

Common Ratio

The common ratio is the constant factor between consecutive terms in a geometric progression. It determines how the sequence behaves, whether it grows, decays, or oscillates, and plays a key role in calculating sums and understanding convergence.

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