What does it mean if the ratio of a geometric sequence is greater than one?

What does it mean if the ratio of a geometric sequence is greater than one?

• A. The series is convergent
• B. The terms decrease
• C. The series diverges
• D. The series remains constant

Answer: (C)

In a geometric sequence, each term is obtained by multiplying the previous term by a fixed ratio. When the ratio of the sequence is greater than one, it indicates that each subsequent term is larger than the previous term. Therefore, as you progress through the terms, the values continue to increase without bound. This characteristic leads to the conclusion that the series diverges. A divergent series means that the sum of the terms does not approach a finite limit; instead, it tends towards infinity. Understanding this helps clarify why the other options do not apply. For instance, a series being convergent would imply that the terms approach a specific limit, which contradicts the growth of the terms in this scenario. Similarly, if the terms were decreasing, the ratio would need to be less than one, leading to terms that converge towards zero or a finite number rather than diverging. Lastly, if the series were to remain constant, the ratio would have to be equal to one, as constant terms do not grow or shrink.

In a geometric sequence, each term is obtained by multiplying the previous term by a fixed ratio. When the ratio of the sequence is greater than one, it indicates that each subsequent term is larger than the previous term. Therefore, as you progress through the terms, the values continue to increase without bound.

This characteristic leads to the conclusion that the series diverges. A divergent series means that the sum of the terms does not approach a finite limit; instead, it tends towards infinity.

Understanding this helps clarify why the other options do not apply. For instance, a series being convergent would imply that the terms approach a specific limit, which contradicts the growth of the terms in this scenario. Similarly, if the terms were decreasing, the ratio would need to be less than one, leading to terms that converge towards zero or a finite number rather than diverging. Lastly, if the series were to remain constant, the ratio would have to be equal to one, as constant terms do not grow or shrink.