Question

when is a sequence geometric?
○ when each term is multiplied by the same number to get the next term
○ when there is a common difference between terms
○ when the sequence increases or decreases in a linear pattern
○ when the ratios between consecutive terms are reciprocals

Explanation:

To determine when a sequence is geometric, recall the definition: A geometric sequence is a sequence where each term after the first is found by multiplying the previous term by a constant called the common ratio (\(r\)).

• Analyze the first option: "when each term is multiplied by the same number to get the next term" matches the definition of a geometric sequence (this "same number" is the common ratio).
• Analyze the second option: A common difference between terms defines an arithmetic sequence, not geometric.
• Analyze the third option: A linear pattern of increase/decrease is characteristic of arithmetic sequences (since the difference is constant), not geometric.
• Analyze the fourth option: If ratios between consecutive terms were reciprocals, the common ratio would not be constant (e.g., if one ratio is \(2\) and the next is \(\frac{1}{2}\), it's not geometric), so this is incorrect.

Answer:

A. when each term is multiplied by the same number to get the next term