Identify a pattern in the given list of numbers. Then use this pattern to find the next number.​ (More than one pattern might​ exist, so it is possible that there is more than one correct​ answer.)
6, 24, 96, 384, 1536

The given sequence is:

\[ 6, 24, 96, 384, 1536 \]

### Pattern 1: Geometric Sequence

Let's examine the relationship between consecutive terms:

1. **First Term to Second Term**: \( 24 \div 6 = 4 \)
2. **Second Term to Third Term**: \( 96 \div 24 = 4 \)
3. **Third Term to Fourth Term**: \( 384 \div 96 = 4 \)
4. **Fourth Term to Fifth Term**: \( 1536 \div 384 = 4 \)

Each term is obtained by multiplying the previous term by 4. Therefore, the pattern here is a geometric sequence with a common ratio of 4.

To find the next number in the sequence:

\[ 1536 \times 4 = 6144 \]

So, the next number in the sequence is **6144**.

### Pattern 2: Exponential Growth with an Increasing Base

Alternatively, observe the following:

- \( 6 = 2^1 \times 3 \)
- \( 24 = 2^3 \times 3 \)
- \( 96 = 2^5 \times 3 \)
- \( 384 = 2^7 \times 3 \)
- \( 1536 = 2^9 \times 3 \)

Here, the power of 2 increases by 2 with each term, while the factor of 3 remains constant.

To find the next number in the sequence:

\[ 2^{11} \times 3 = 2048 \times 3 = 6144 \]

This also gives the next number as **6144**.

### Conclusion

Both patterns lead to the same next number, **6144**.

Would you like further details on geometric sequences or exponential growth, or do you have any other questions?

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Here are 5 relative questions to consider:

1. What would be the 10th term in the sequence using the geometric pattern?
2. How can the general formula for the \(n\)th term be derived for both patterns?
3. What is the sum of the first 6 terms of this sequence?
4. How does the common ratio or base exponent affect the sequence's growth rate?
5. Can you create a sequence using a similar pattern but with a different initial number?

**Tip:** Identifying multiple patterns in a sequence can reveal deeper insights into its structure, offering more than one correct next term in some cases.

Explore how to identify and extend patterns in sequences using geometric sequences and exponential growth. This example sequence, 6, 24, 96, 384, 1536, is analyzed to find the next number using multiple patterns. Learn about geometric sequences with a common ratio and exponential growth with increasing base. Suitable for high school students.

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