Ohio Assessments for Educators (OAE) Mathematics Practice Exam

Which property defines the behavior of a geometric sequence?

1. The common difference remains constant

2. The ratio between consecutive terms remains constant

3. The sum of the terms is fixed

4. The terms alternate in sign

The correct answer is: The ratio between consecutive terms remains constant

A geometric sequence is characterized by the property that the ratio between consecutive terms remains constant. This means that if you take any term in the sequence and divide it by the term that precedes it, the result will be the same throughout the sequence. This constant ratio is what differentiates a geometric sequence from other types of sequences, such as arithmetic sequences, where a fixed common difference is maintained instead. In a geometric sequence, if the first term is \( a \) and the common ratio is \( r \), the terms can be expressed as \( a, ar, ar^2, ar^3, \ldots \). This demonstrates that each term is obtained by multiplying the previous term by the same constant \( r \). This fundamental property of constant ratios allows for various analyses and applications in mathematics, including scenarios involving exponential growth or decay, which are commonly modeled using geometric sequences. The other options do not accurately describe geometric sequences: the common difference refers to arithmetic sequences, the sum of the terms is not fixed as it depends on both the number of terms and the common ratio, and alternating signs relate more to specific sequences rather than defining the general behavior of a geometric sequence.