12. Write the formula for the nth term of the geometric sequence: 3, 12, 48,

The formula for the nth term of a geometric sequence can be written as:

\[
a_n = a_1 \cdot r^{(n-1)}
\]

where:
- \(a_n\) is the nth term,
- \(a_1\) is the first term,
- \(r\) is the common ratio, and
- \(n\) is the term number.

For the given sequence: 3, 12, 48, ...

1. The first term \(a_1 = 3\),
2. To find the common ratio \(r\), divide the second term by the first term: 

\[
r = \frac{12}{3} = 4
\]

Thus, the common ratio is \(r = 4\).

Therefore, the formula for the nth term is:

\[
a_n = 3 \cdot 4^{(n-1)}
\]

Would you like more details on how geometric sequences work or have any other questions?

Here are 5 related questions:
1. How do you find the sum of the first n terms of a geometric sequence?
2. What is the difference between an arithmetic and geometric sequence?
3. Can a geometric sequence have a negative common ratio?
4. How do you determine if a sequence is geometric?
5. How would the nth term formula change if the common ratio is a fraction?

**Tip:** In geometric sequences, if the common ratio \(r > 1\), the terms grow quickly; if \(r\) is between 0 and 1, the terms shrink.

Write the formula for the nth term of the geometric sequence: 3, 12, 48.

Learn how to derive the formula for the nth term of a geometric sequence, with the sequence 3, 12, 48 as an example. This solution includes key concepts like geometric sequences, common ratios, and the exponential growth formula. Suitable for students in Grades 8-10.

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