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 $|2008|$
(A) $-4$
(B) $-12$
(C) 12
(D) 4

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Key Concepts

Absolute Value

The absolute value of a number represents its non-negative magnitude regardless of its sign. This concept is important when the problem requires a focus on the size or magnitude of a term without considering whether it is positive or negative.

Systems of Equations

Problems involving sums of certain terms in a geometric progression often lead to the formation of systems of equations. Solving these systems enables one to find unknown values such as the first term or the common ratio by relating different parts of the progression.

Alternating Signs

When the terms of a sequence are alternately positive and negative, it indicates that the common ratio in the geometric progression is negative. This insight helps set up the correct relationships between terms, vital for solving problems involving sign patterns.

Common Ratio

The common ratio is the constant factor between consecutive terms in a geometric progression. Its value influences the growth or decay of the sequence, and if it is negative, the signs of the terms will alternate.

Geometric Progression

A geometric progression is a sequence where each term is derived by multiplying the previous term by a constant factor called the common ratio. Understanding this concept is essential to relate different terms in the sequence through multiplication.

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