Which of the following is a characteristic of a geometric sequence?

• A. Each term is the sum of the two preceding terms
• B. Each term is found by multiplying the previous term by a constant
• C. Each term is the average of its neighbors
• D. Each term alternates in sign

Answer: (B)

A geometric sequence is defined by the property that each term after the first is generated by multiplying the previous term by a constant, known as the common ratio. This means that if you have the first term (let's say ( a )) and the common ratio (let's say ( r )), the terms of the sequence can be represented as ( a, ar, ar^2, ar^3, ) and so forth. This consistent multiplicative relationship is the hallmark of geometric sequences and distinguishes them from other types of sequences.

In contrast, the other options describe characteristics of different types of sequences. For example, one option suggests that each term is the sum of the two preceding terms, which describes a Fibonacci-like sequence and is the defining property of a recursive sequence rather than a geometric one. Another option, stating that each term is the average of its neighbors, refers to an averaging sequence, which again does not align with the properties of a geometric sequence. Lastly, the option regarding alternating signs could describe specific types of sequences but does not capture the essence of a geometric sequence. Thus, the correct identification of option B underscores the fundamental definition of geometric sequences in mathematics.