To find the missing number in the sequence \(12, 15, 75, ?, 738, 748\), let's first analyze the pattern:

1. **First Difference Analysis**:
   - \(15 - 12 = 3\)
   - \(75 - 15 = 60\)

   There is no immediate simple arithmetic progression in the first differences.

2. **Second Difference Analysis**:
   - \(60 - 3 = 57\)

   Still no clear arithmetic progression.

3. **Factor and Ratio Analysis**:
   - \(15 / 12 = 1.25\)
   - \(75 / 15 = 5\)

   The ratios are also not forming a consistent pattern.

4. **Identifying a Pattern**:
   Let's consider multiplication and addition patterns:
   - From \(12\) to \(15\): \(12 \times 1.25 = 15\)
   - From \(15\) to \(75\): \(15 \times 5 = 75\)

   The next number could be calculated similarly using a simple multiplicative pattern or a different approach.

5. **Possible Step to Identify Next Number**:
   The jump from 75 to 738 is large. To account for this, let's consider an additive factor:
   - \(738 - 75 = 663\)

   This could suggest the sequence follows a compound pattern.

Given these steps, if we examine sequences where we increase by a multiplicative factor or large jump, one possibility is the next number could follow by adding a significant constant or involve a complex multiplication.

Since the jump from 75 to 738 is significant, we hypothesize the next number could involve an addition operation (e.g., \(75 + 663 = 738\)).

**Next Potential Number**:
To continue the sequence:
- Adding a smaller number as we get closer to 738: \(75 + 250 = 325\) could be a simpler pattern.
- Or a multiplicative factor close to \(9.84\) or similar large constants.

However, without a clear simple mathematical operation, the missing number in a complex or seemingly non-linear sequence can often be ambiguous without further pattern evidence.

**Assumed Missing Number**: **325** (hypothesized based on a possible arithmetic addition pattern).

Would you like to go into more details on different possible sequence patterns, or how different methods of sequence finding could apply?

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Here are 5 follow-up questions:
1. What if the sequence followed a quadratic or higher polynomial pattern?
2. Could there be a combined additive and multiplicative pattern?
3. What other types of sequence patterns are common in advanced mathematical problems?
4. How would the sequence change if we applied exponential growth or decay?
5. Is there a method for systematically determining the pattern in mixed-type sequences?

**Tip**: When faced with complex sequences, try various pattern recognitions, such as differences, ratios, or even considering polynomial relationships.

Explore the solution to a sequence problem involving arithmetic and multiplicative patterns. Learn how to identify complex sequence patterns using mathematical analysis and hypothesis testing. Suitable for advanced mathematics enthusiasts.

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