To classify the given set of numbers:
3, 1, 1/3, 1/9, 1/27, ...
We should first analyze the relationship between the consecutive terms.
- Determine the ratio between consecutive terms:
- From 3 to 1: \( \frac{1}{3} \)
- From 1 to \( \frac{1}{3} \): \( \frac{1/3}{1} = \frac{1}{3} \)
- From \( \frac{1}{3} \) to \( \frac{1}{9} \): \( \frac{1/9}{1/3} = \frac{1/9 \cdot 3}{1} = \frac{1}{3} \)
- From \( \frac{1}{9} \) to \( \frac{1}{27} \): \( \frac{1/27}{1/9} = \frac{1/27 \cdot 9}{1} = \frac{1}{3} \)
In each case, we see that each term is obtained by multiplying the previous term by \( \frac{1}{3} \).
- Definition check:
- An arithmetic sequence involves adding a constant difference to each term (e.g., 2, 4, 6).
- A geometric sequence involves multiplying each term by a constant ratio (e.g., 2, 4, 8).
Since the ratio between each term is constant (\( \frac{1}{3} \)), we find that this set is a geometric sequence.
Thus, the correct option is: Option #2: The set is a geometric sequence.