The sequence shown in the image is:

$0, 1, 1, 2, 4, 7, 13, 24, 44, \_$

To solve for the next number in the sequence, let's analyze the pattern:

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Step 1: Find the relationship between terms.

• Look at the differences between consecutive terms: $1 - 0 = 1, \quad 1 - 1 = 0, \quad 2 - 1 = 1, \quad 4 - 2 = 2, \quad 7 - 4 = 3, \quad 13 - 7 = 6, \quad 24 - 13 = 11, \quad 44 - 24 = 20$
• Differences: $1, 0, 1, 2, 3, 6, 11, 20$

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Step 2: Analyze the second differences.

• Look at the differences of the differences: $0 - 1 = -1, \quad 1 - 0 = 1, \quad 2 - 1 = 1, \quad 3 - 2 = 1, \quad 6 - 3 = 3, \quad 11 - 6 = 5, \quad 20 - 11 = 9$
• Second differences: $-1, 1, 1, 1, 3, 5, 9$

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Step 3: Predict the next term in the second differences.

• The second differences themselves are not constant, but they follow a noticeable pattern. Observe: $1, 3, 5, 9 \quad \text{(increasing unevenly)}.$ The difference between these terms is: $3 - 1 = 2, \quad 5 - 3 = 2, \quad 9 - 5 = 4$ Thus, the next difference is likely $4 + 2 = 6$, so the next second difference should be $9 + 6 = 15$.

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Step 4: Use this to find the next difference in the sequence.

• Add $15$ to the last first difference: $20 + 15 = 35$

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Step 5: Calculate the next term in the original sequence.

• Add $35$ to the last term ($44$): $44 + 35 = 79$

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Final Answer:

The next number in the sequence is: $\boxed{79}$

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Would you like further clarification or a deeper exploration of this sequence?

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5 Related Questions:

1. How can you verify if a sequence follows a Fibonacci-like structure?
2. What are second differences, and how are they used in sequences?
3. Can you derive a formula for this specific sequence?
4. What other techniques are used to identify patterns in numerical sequences?
5. How are differences and sums used in solving real-world mathematical problems?

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Tip:

When solving number sequences, always start by looking at consecutive differences, as patterns often emerge from these calculations!