The sequence given in the problem is:

3, 12, 48, 192, ...

Identifying the pattern:

• The ratio between consecutive terms is constant:
  $\frac{12}{3} = 4$,
  $\frac{48}{12} = 4$,
  $\frac{192}{48} = 4$.

This shows that the sequence is geometric, with a common ratio $r = 4$.

1. Explicit Formula:

For a geometric sequence, the explicit formula is given by:

$a_n = a_1 \cdot r^{n-1}$

Where:

• $a_1$ is the first term,
• $r$ is the common ratio, and
• $n$ is the term number.

For this sequence:

• $a_1 = 3$,
• $r = 4$.

Thus, the explicit formula is:

$a_n = 3 \cdot 4^{n-1}$

2. Recursive Formula:

For a geometric sequence, the recursive formula is written as:

$a_1 = 3$ $a_n = 4 \cdot a_{n-1}, \, \text{for} \, n > 1$

In summary:

• Explicit: $a_n = 3 \cdot 4^{n-1}$,
• Recursive: $a_1 = 3, \, a_n = 4 \cdot a_{n-1}$.

Would you like further details or have any questions?

Here are 5 related questions to expand your understanding:

1. How does the common ratio affect the growth of a geometric sequence?
2. What is the difference between an arithmetic and a geometric sequence?
3. How would you find the 10th term of this sequence using the explicit formula?
4. Can a geometric sequence have a common ratio less than 1?
5. How can the recursive formula be used to compute higher terms efficiently?

Tip: Always check for common differences (arithmetic) or ratios (geometric) when identifying sequences!