To find the replacement for $a$ and $b$ in the sequence $8748, 2916, a, 324, b, 36, 12$, we need to identify the pattern in the sequence.

Let's examine the relationship between consecutive terms:

1. $8748$ to $2916$: $\frac{8748}{2916} = 3$ So, $2916 = \frac{8748}{3}$.

2. $2916$ to $a$: $a = \frac{2916}{3} = 972$

3. $a$ to $324$: $\frac{972}{324} = 3$ So, $324 = \frac{972}{3}$.

4. $324$ to $b$: $b = \frac{324}{3} = 108$

5. $b$ to $36$: $\frac{108}{36} = 3$ So, $36 = \frac{108}{3}$.

6. $36$ to $12$: $\frac{36}{12} = 3$ So, $12 = \frac{36}{3}$.

Based on this pattern, every term is $\frac{1}{3}$ of the previous term. Thus, the sequence with the replacements for $a$ and $b$ is:

$8748, 2916, 972, 324, 108, 36, 12$

So, the values for $a$ and $b$ are:

• $a = 972$
• $b = 108$

Would you like more details or have any questions?

Here are 8 related questions to consider:

1. How can the general formula for this sequence be written?
2. What are other types of common mathematical sequences?
3. How does identifying patterns in sequences help in solving real-world problems?
4. What is the importance of recognizing geometric sequences in algebra?
5. Can you provide an example of an arithmetic sequence?
6. How do you find the $n$-th term of a geometric sequence?
7. What are some practical applications of geometric sequences?
8. How do sequences relate to series in mathematics?

Tip: When working with sequences, look for common ratios or differences to identify whether they are geometric or arithmetic sequences.