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

a) 2, 14, 98, 686,


Let's analyze the sequence: 2, 14, 98, 686.

### Step 1: Check for a common difference (Arithmetic Sequence)
An arithmetic sequence has a constant difference between consecutive terms.

- \(14 - 2 = 12\)
- \(98 - 14 = 84\)
- \(686 - 98 = 588\)

Since the differences between consecutive terms are not the same, this sequence is **not arithmetic**.

### Step 2: Check for a common ratio (Geometric Sequence)
A geometric sequence has a constant ratio between consecutive terms. To check if the sequence is geometric, divide each term by the previous term:

- \( \frac{14}{2} = 7 \)
- \( \frac{98}{14} = 7 \)
- \( \frac{686}{98} = 7 \)

Since the ratio is constant (\(r = 7\)), this is a **geometric sequence** with a common ratio of \(r = 7\).

### Conclusion:
The sequence \(2, 14, 98, 686\) is a geometric sequence with a common ratio of \(7\).

Would you like more details or have any questions?

Here are some related questions:
1. What is the general formula for a geometric sequence?
2. How can we find the sum of a finite geometric sequence?
3. Can a sequence be both arithmetic and geometric?
4. What is the difference between an arithmetic and a geometric sequence?
5. How does a geometric sequence differ from a harmonic sequence?

**Tip:** In a geometric sequence, the ratio between consecutive terms is always the same, and this is called the common ratio.

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

a) 2, 14, 98, 686

Analyze the sequence 2, 14, 98, 686 to determine if it's arithmetic or geometric. Learn how to check for common differences or ratios, and find out why this sequence is geometric with a common ratio of 7. Suitable for students in Grades 6-8.

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