Let's examine the sequence $8, 12, 8, 4, 0$ to determine whether it is arithmetic, geometric, or otherwise.

Step 1: Check if the sequence is arithmetic

In an arithmetic sequence, the difference between consecutive terms is constant. Let's find the differences between consecutive terms:

• $12 - 8 = 4$
• $8 - 12 = -4$
• $4 - 8 = -4$
• $0 - 4 = -4$

Since the difference is not consistent throughout (it changes from 4 to -4), this is not an arithmetic sequence.

Step 2: Check if the sequence is geometric

In a geometric sequence, the ratio between consecutive terms is constant. Let's find the ratios between consecutive terms:

• $\frac{12}{8} = 1.5$
• $\frac{8}{12} = \frac{2}{3}$
• $\frac{4}{8} = 0.5$
• $\frac{0}{4} = 0$

Since the ratio is not constant, this is not a geometric sequence.

Conclusion:

The sequence $8, 12, 8, 4, 0$ is neither arithmetic nor geometric. Therefore, it falls into the "otherwise" category, meaning it doesn't follow the rules for either type of sequence.

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Would you like further explanation or have any specific questions?

Here are 5 related questions to consider:

1. What is the general form of an arithmetic sequence?
2. What conditions must be met for a sequence to be geometric?
3. How can a recursive formula help define a non-arithmetic, non-geometric sequence?
4. Can a sequence be both arithmetic and geometric?
5. How would the sequence change if the first few terms were modified?

Tip: When identifying the type of sequence, always calculate both the difference between terms and the ratio of terms to check for consistency.