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.