The sequence provided: 2, 8, 32, 128 follows a geometric pattern.

Here's the reasoning:
- The first term is 2.
- The second term is 8, which is obtained by multiplying 2 by 4.
- The third term is 32, which is obtained by multiplying 8 by 4.
- The fourth term is 128, which is obtained by multiplying 32 by 4.

This is a geometric sequence where the common ratio \( r = 4 \).

### General Form of a Geometric Sequence:
For a geometric sequence, the \( n \)-th term can be expressed as:
\[
a_n = a_1 \cdot r^{(n-1)}
\]
where:
- \( a_1 \) is the first term,
- \( r \) is the common ratio,
- \( n \) is the position of the term in the sequence.

In this case:
- \( a_1 = 2 \),
- \( r = 4 \).

So, the general formula for the \( n \)-th term in this sequence is:
\[
a_n = 2 \cdot 4^{(n-1)}
\]

For example:
- The 5th term would be:
\[
a_5 = 2 \cdot 4^{(5-1)} = 2 \cdot 4^4 = 2 \cdot 256 = 512
\]

Would you like to know more details or have any questions?

### 5 Related Questions:
1. How can you determine if a sequence is arithmetic or geometric?
2. What is the sum of the first \( n \) terms of a geometric series?
3. How do you find the common ratio in a geometric sequence?
4. Can geometric sequences have negative or fractional ratios?
5. How does the behavior of a geometric sequence change if the common ratio is less than 1?

**Tip:** In a geometric sequence, multiplying by a common ratio greater than 1 results in exponential growth, while a ratio between 0 and 1 leads to exponential decay.

Explore the geometric sequence 2, 8, 32, 128 and learn how to identify the common ratio and find the general formula for any term in the sequence. This guide provides a step-by-step solution, suitable for students in Grades 8-10.

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