Find the next number in the sequence 1,4,11,29,76,199

To find the next number in the sequence \(1, 4, 11, 29, 76, 199\), let's analyze the pattern by finding the differences between consecutive terms.

1. Difference between \(4\) and \(1\): \(4 - 1 = 3\)
2. Difference between \(11\) and \(4\): \(11 - 4 = 7\)
3. Difference between \(29\) and \(11\): \(29 - 11 = 18\)
4. Difference between \(76\) and \(29\): \(76 - 29 = 47\)
5. Difference between \(199\) and \(76\): \(199 - 76 = 123\)

The differences between the terms are \(3, 7, 18, 47, 123\).

Now, let's look at the differences between these differences:

1. Difference between \(7\) and \(3\): \(7 - 3 = 4\)
2. Difference between \(18\) and \(7\): \(18 - 7 = 11\)
3. Difference between \(47\) and \(18\): \(47 - 18 = 29\)
4. Difference between \(123\) and \(47\): \(123 - 47 = 76\)

The second differences are \(4, 11, 29, 76\), which appear to be following the original sequence. This suggests that the sequence of differences is recursive.

Let's assume that the next difference will be:

\[ 76 + 199 = 275 \]

Now, to find the next number in the sequence, add this difference to the last number:

\[ 199 + 275 = 474 \]

So, the next number in the sequence is **474**.

Would you like more details or have any questions?

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Here are some related questions:

1. What is the formula for finding the \(n\)th term in this sequence?
2. How do you determine the general pattern in a sequence?
3. What are some common types of sequences in mathematics?
4. Can you find the next term in the sequence \(2, 5, 10, 17, 26\)?
5. How do you find the sum of an arithmetic sequence?
6. What is the difference between arithmetic and geometric sequences?
7. How can recursive sequences be used in real-life problems?
8. Can you provide an example of a sequence that does not follow a simple pattern?

**Tip:** When analyzing a sequence, always check the first and second differences to identify potential patterns.

Discover how to find the next number in the sequence 1, 4, 11, 29, 76, 199 with a step-by-step analysis. Understand the pattern of differences and recursive sequences in mathematics. Suitable for advanced high school students and anyone interested in sequence analysis.

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