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.