3) Determine whether each sequence is arithmetic or geometricor otherwise. If it is a geometric sequence, find the common ratio.

b) 8. 12, 8, 4, 0,

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.

Determine whether each sequence is arithmetic or geometric or otherwise. If it is a geometric sequence, find the common ratio. b) 8, 12, 8, 4, 0

Explore how to determine if a sequence is arithmetic, geometric, or neither, with a step-by-step analysis of the sequence 8, 12, 8, 4, 0. This problem covers core mathematical concepts such as differences and ratios, suitable for students in Grades 6-8.

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