Good evening! ♥️ Let's work through these logical questions step by step.

1. What numbers are next?

Sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, _?, _?, _?.

Solution: This sequence is the Fibonacci sequence, where each number is the sum of the two preceding ones.

• $0 + 1 = 1$
• $1 + 1 = 2$
• $1 + 2 = 3$
• $2 + 3 = 5$
• $3 + 5 = 8$
• $5 + 8 = 13$
• $8 + 13 = 21$

Continuing the sequence:

• $13 + 21 = 34$
• $21 + 34 = 55$
• $34 + 55 = 89$

Answer: 34, 55, 89

2. What would be the next?

Sequence: 78, 91, 169, 260, _?.

Solution: This sequence isn't as immediately recognizable, so let's look for a pattern:

• The difference between 91 and 78 is 13.
• The difference between 169 and 91 is 78.
• The difference between 260 and 169 is 91.

The differences form their own sequence: 13, 78, 91. Notice that 78 and 91 are consecutive numbers from the first sequence. Let's add 169 (which is the third term in the original sequence) to the last number in the difference sequence:

• Next difference = 260 + 169 = 429

Answer: 429

3. What would be the first 2 consecutive numbers in the items below?

Sequence: _?, _?, 233, 377, 610.

Solution: This sequence appears to be a continuation of the Fibonacci sequence starting from 233. We know:

• 233, 377, 610

In the Fibonacci sequence, the numbers before 233 are:

• $233 - 144 = 89$ (the previous term)
• $144 - 89 = 55$ (the term before that)

Answer: 89, 144

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Would you like more details or explanations? Here are some related questions to deepen your understanding:

1. How is the Fibonacci sequence applied in nature?
2. Can you find the 15th term in the Fibonacci sequence?
3. What is the formula for finding the $n$th Fibonacci number?
4. What are some other famous mathematical sequences?
5. How can you identify the pattern in a given sequence?

Tip: When dealing with sequences, always check for common differences or ratios. They often reveal the pattern!