Question

Classify the sequence left-brace a subscript n baseline right-brace equals left-brace 3 comma negative 1 comma Start Fraction 1 over 3 End Fraction comma negative Start Fraction 1 over 9 End Fraction ellipsis right-brace as arithmetic, geometric, or neither. If there is not enough information to classify the sequence, choose not enough information.
(1 point)
Responses

arithmetic
arithmetic

geometric
geometric

neither
neither

not enough information

Answer 1

To classify the sequence \( \{ a_n \} = \{ 3, -1, \frac{1}{3}, -\frac{1}{9}, \ldots \} \), we need to determine if it is arithmetic, geometric, or neither.

1. **Arithmetic Sequence**: A sequence is arithmetic if the difference between consecutive terms is constant. Let's calculate the differences:
- \( -1 - 3 = -4 \)
- \( \frac{1}{3} - (-1) = \frac{1}{3} + 1 = \frac{4}{3} \)
- \( -\frac{1}{9} - \frac{1}{3} = -\frac{1}{9} - \frac{3}{9} = -\frac{4}{9} \)

The differences are -4, \( \frac{4}{3} \), and \( -\frac{4}{9} \), which are not constant. Therefore, it is not an arithmetic sequence.

2. **Geometric Sequence**: A sequence is geometric if the ratio between consecutive terms is constant. Let's calculate the ratios:
- \( \frac{-1}{3} \)
- \( \frac{\frac{1}{3}}{-1} = -\frac{1}{3} \)
- \( \frac{-\frac{1}{9}}{\frac{1}{3}} = -\frac{1}{9} \times \frac{3}{1} = -\frac{1}{3} \)

The ratios \( \frac{-1}{3} \), and \( \frac{-1}{3} \), and \( \frac{-1}{3} \) are constant.

Since the sequence has a constant ratio, it is classified as a **geometric sequence**.

The answer is:
**geometric**