Determine if the following sequence is arithmetic, geometric, or neither. Explain.

2, 8, 26, 80, . . .

Group of answer choices

Arithmetic; The common difference is 6.

Geometric; The common ratio is 6.

Both; There is a common ratio of 3 and common difference of 2.

Neither; There is no common ratio and no common difference.

Geometric; The common ratio is 4.

To determine whether the sequence \( 2, 8, 26, 80, \dots \) is arithmetic, geometric, or neither, we need to check if there is a common difference (for arithmetic) or a common ratio (for geometric).

### Checking for an Arithmetic Sequence:

An **arithmetic sequence** has a constant **difference** between consecutive terms. We calculate the differences between consecutive terms:

1. \( 8 - 2 = 6 \)
2. \( 26 - 8 = 18 \)
3. \( 80 - 26 = 54 \)

Since the differences are not the same (\( 6, 18, 54 \)), this is **not an arithmetic sequence**.

### Checking for a Geometric Sequence:

A **geometric sequence** has a constant **ratio** between consecutive terms. We calculate the ratios between consecutive terms:

1. \( \frac{8}{2} = 4 \)
2. \( \frac{26}{8} = 3.25 \)
3. \( \frac{80}{26} \approx 3.08 \)

Since the ratios are not the same (\( 4, 3.25, 3.08 \)), this is **not a geometric sequence**.

### Conclusion:
Since there is no common difference or common ratio, the sequence is **neither arithmetic nor geometric**.

Thus, the correct answer is:

**Neither; There is no common ratio and no common difference.**

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Do you want more details or have any questions?

Here are 5 related questions to expand on this topic:
1. What is the formula for the \(n\)-th term of an arithmetic sequence?
2. How do you find the common ratio of a geometric sequence?
3. Can a sequence be both arithmetic and geometric?
4. What is the difference between an arithmetic progression and an arithmetic series?
5. How do you determine if a sequence is quadratic?

**Tip:** When working with sequences, always try to check both for common differences and common ratios to classify them correctly.

Determine if the following sequence is arithmetic, geometric, or neither. Explain. 2, 8, 26, 80, . . . Group of answer choices: Arithmetic; The common difference is 6. Geometric; The common ratio is 6. Both; There is a common ratio of 3 and common difference of 2. Neither; There is no common ratio and no common difference. Geometric; The common ratio is 4.

Analyze the sequence 2, 8, 26, 80 to determine if it is arithmetic, geometric, or neither. Learn how to check for common differences and common ratios to classify sequences. This problem covers key mathematical concepts related to sequences and series, suitable for Grades 9-11.

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