Which statement describes the behavior of a geometric sequence?

• A. It approaches the value of zero as it progresses
• B. Each term is a sum of all preceding terms
• C. Each term is a fixed multiple of the previous term
• D. It alternates in sign

Answer: (C)

The behavior of a geometric sequence is characterized by each term being a fixed multiple of the previous term. In a geometric sequence, this constant ratio between consecutive terms defines how the sequence progresses. For instance, if the first term is ( a ) and the common ratio is ( r ), the terms can be expressed as ( a, ar, ar^2, ar^3, ) and so on. This formula illustrates that each subsequent term results from multiplying the previous term by ( r ), thereby emphasizing the multiplicative relationship central to geometric sequences.

This property distinguishes geometric sequences from others, such as arithmetic sequences, where each term is obtained by adding a constant value to the previous term, or other behavior patterns that do not rely on a fixed ratio. Understanding this characteristic can help in recognizing and formulating geometric sequences in various mathematical contexts.