https://file.aiquickdraw.com/mathbot/file/1725982114102tew7hi7s.jpg

Let's analyze each sequence from the image to determine if it is geometric, and if so, find the common ratio \( r \).

### (a) Sequence: \( -4, -20, -100, -500, \dots \)

To check if a sequence is geometric, divide each term by the previous term and see if the ratio is constant.

\[
r_1 = \frac{-20}{-4} = 5, \quad r_2 = \frac{-100}{-20} = 5, \quad r_3 = \frac{-500}{-100} = 5
\]

Since the ratio is constant (\( r = 5 \)), the sequence is geometric.

### (b) Sequence: \( 24, 20, 16, 12, \dots \)

Now, let's check the common ratio for the second sequence:

\[
r_1 = \frac{20}{24} = \frac{5}{6}, \quad r_2 = \frac{16}{20} = \frac{4}{5}, \quad r_3 = \frac{12}{16} = \frac{3}{4}
\]

The ratios are not the same, so the sequence is **not geometric**.

### (c) Sequence: \( 256, 64, 16, 4, \dots \)

Checking the common ratio for the third sequence:

\[
r_1 = \frac{64}{256} = \frac{1}{4}, \quad r_2 = \frac{16}{64} = \frac{1}{4}, \quad r_3 = \frac{4}{16} = \frac{1}{4}
\]

Since the ratio is constant (\( r = \frac{1}{4} \)), the sequence is geometric.

### Summary:
- **(a)** Geometric with common ratio \( r = 5 \)
- **(b)** Not geometric
- **(c)** Geometric with common ratio \( r = \frac{1}{4} \)

Do you want more details or have any questions? Here are some related questions to consider:

1. How can you recognize a geometric sequence without calculating every ratio?
2. What are the differences between arithmetic and geometric sequences?
3. Can a geometric sequence have a common ratio of 1? What would it look like?
4. How do you find the general formula for the \( n \)-th term of a geometric sequence?
5. Can a geometric sequence have negative or fractional common ratios?

**Tip:** In a geometric sequence, the common ratio \( r \) can be found by dividing any term by its previous term. This constant \( r \) defines the relationship between consecutive terms.

For each sequence, determine whether it appears to be geometric. If it does, find the common ratio.

Determine whether sequences are geometric and find the common ratio. This task covers the geometric sequence formula, including step-by-step checks for constant ratios between terms. Suitable for high school students learning about sequences.

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