Question

For each sequence, determine whether it appears to be geometric.
If it does, find the common ratio.
(a) 10, 30, 90, 270, ...
• Geometric
Common ratio:
r =
• Not geometric
(b) 10, 15, 20, 25, ...
• Geometric
Common ratio:
r =
• Not geometric
(c) 250, - 50, 10, - 2, ...
• Geometric
Common ratio:
• Not geometric

Answer 1

To determine whether a sequence is geometric, we need to check if the ratio of consecutive terms is constant throughout the sequence. Let's analyze each sequence:

(a) **10, 30, 90, 270, ...**
- To find the common ratio \( r \):
- \( r = \frac{30}{10} = 3 \)
- \( r = \frac{90}{30} = 3 \)
- \( r = \frac{270}{90} = 3 \)

Since the ratio is constant (3), the sequence is geometric.

**Geometric**
**Common ratio**: \( r = 3 \)

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(b) **10, 15, 20, 25, ...**
- To find the common ratio \( r \):
- \( r = \frac{15}{10} = 1.5 \)
- \( r = \frac{20}{15} = \frac{4}{3} \approx 1.333 \)
- \( r = \frac{25}{20} = 1.25 \)

Since the ratios are not constant (they vary), the sequence is not geometric.

**Not geometric**

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(c) **250, -50, 10, -2, ...**
- To find the common ratio \( r \):
- \( r = \frac{-50}{250} = -0.2 \)
- \( r = \frac{10}{-50} = -0.2 \)
- \( r = \frac{-2}{10} = -0.2 \)

Since the ratio is constant (-0.2), the sequence is geometric.

**Geometric**
**Common ratio**: \( r = -0.2 \)

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To summarize:
(a) Geometric, \( r = 3 \)
(b) Not geometric
(c) Geometric, \( r = -0.2 \)