https://file.aiquickdraw.com/mathbot/file/17268584265918c8piwxp.jpg

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**: \( a_1 = 3, \, a_n = 4 \cdot a_{n-1} \).

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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!

Write the explicit and recursive equations for the sequence: 3, 12, 48, 192, ...

Learn how to find the explicit and recursive formulas for a geometric sequence with the first few terms 3, 12, 48, 192. This problem involves key algebraic concepts like geometric progression and recursive sequences, suitable for Grades 8-10.

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