Question

For each sequence, determine whether it appears to be geometric.
If it does, find the common ratie.
(a) -2, -8, -32, -128, ...
• Geometric
Common ratio: r = _
• Not geometric
(b) 12, 10, 8, 6, ...
• Geometric
Common ratio: r = _
• Not geometric
(c)
250, 50, 10, 2, ...
• Geometric
Common ratio: r = _
• Not geometric

Answer 1

To determine whether each sequence is geometric, we need to check if the ratio of successive terms is constant. A geometric sequence has a common ratio \( r \) that can be calculated by dividing any term by the preceding term.

Let’s evaluate each sequence:

(a) **Sequence: -2, -8, -32, -128, ...**
- Common ratio calculation:
- \( r = \frac{-8}{-2} = 4 \)
- \( r = \frac{-32}{-8} = 4 \)
- \( r = \frac{-128}{-32} = 4 \)

Since the ratio is constant, this sequence is geometric.
- **Geometric**
- **Common ratio: \( r = 4 \)**

(b) **Sequence: 12, 10, 8, 6, ...**
- Common ratio calculation:
- \( r = \frac{10}{12} = \frac{5}{6} \)
- \( r = \frac{8}{10} = \frac{4}{5} \)
- \( r = \frac{6}{8} = \frac{3}{4} \)

Since the ratios are not constant, this sequence is not geometric.
- **Not geometric**

(c) **Sequence: 250, 50, 10, 2, ...**
- Common ratio calculation:
- \( r = \frac{50}{250} = \frac{1}{5} \)
- \( r = \frac{10}{50} = \frac{1}{5} \)
- \( r = \frac{2}{10} = \frac{1}{5} \)

Since the ratio is constant, this sequence is geometric.
- **Geometric**
- **Common ratio: \( r = \frac{1}{5} \)**

In summary:
(a) Geometric, common ratio \( r = 4 \)
(b) Not geometric
(c) Geometric, common ratio \( r = \frac{1}{5} \)