To find the equation for the given sequence $3, 12, 48, \ldots$, we need to determine the pattern that describes how each term relates to the previous term.

Step 1: Identify the pattern

Looking at the sequence:

• The first term ($n = 1$) is 3.
• The second term ($n = 2$) is 12.
• The third term ($n = 3$) is 48.

We can see that each term is multiplied by 4 to get the next term:

• $12 = 3 \times 4$
• $48 = 12 \times 4$

Thus, the sequence has a common ratio of 4, making it a geometric sequence.

Step 2: Write the formula for the general term

For a geometric sequence, the general term is given by:

$f(n) = a \cdot r^{(n-1)}$

where:

• $a$ is the first term,
• $r$ is the common ratio.

Step 3: Substitute the values

• The first term $a = 3$.
• The common ratio $r = 4$.

So, the equation for the sequence is:

$f(n) = 3 \cdot 4^{(n-1)}$

Therefore, the boxes should be filled as follows:

• The first box: $3$
• The second box: $4$

Let me know if you would like further details or have any questions!

Here are some related questions you might find useful:

1. How do you find the nth term of an arithmetic sequence?
2. What is the sum of the first n terms of a geometric sequence?
3. How can you determine whether a sequence is arithmetic or geometric?
4. What are the differences between geometric and arithmetic sequences?
5. How do you calculate the common ratio in a geometric sequence?

Tip: When working with sequences, always identify whether the pattern involves addition (arithmetic) or multiplication (geometric) to determine the type of sequence.