The first two terms of a geometric progression add up to 12...

The first two terms of a geometric progression add up to 12 . The sum of the third and the fourth terms is 48 . If the terms of the geometric progression are alternately positive and negative, then the first term is
(A) $-4$
(B) $-12$
(C) 12
(D) 4

Key Concepts

Formulating Equations in Geometric Sequences

Problems involving geometric progressions often require setting up algebraic equations based on the relationships between terms, such as the sum of specific terms. Using the general form of the geometric progression, these equations allow one to solve for unknown quantities like the first term or the common ratio by expressing the sums in terms of these variables.

Alternating Signs

When the terms of a geometric progression alternate in sign, it indicates that the common ratio is negative. This property results in consecutive terms having opposite signs, which is an important consideration when forming and solving equations, as it affects both the sign and magnitude of the calculated terms.

Geometric Progression

A geometric progression is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant factor, known as the common ratio. This type of sequence is characterized by exponential growth or decay, and the relations between the terms are governed by this constant multiplier.

Common Ratio

The common ratio in a geometric progression is the constant multiplier that generates each term from its previous term. It is determined by dividing any term by the term that precedes it. The value and sign of the common ratio strongly influence the behavior of the sequence, including whether the terms increase, decrease, or alternate in sign.

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