https://file.aiquickdraw.com/mathbot/file/17246913584087zl1lvt1.jpg

Determine the pattern and find the missing term in each of the following sequence

I'll analyze the image and determine the pattern to find the missing terms in each sequence.

### 1. Sequence: 2, 5, 10, ___, 26, 37, 50, ___
- **Pattern**: Each term increases by successive odd numbers.
  - 2 + 3 = 5
  - 5 + 5 = 10
  - 10 + 7 = 17
  - 17 + 9 = 26
  - 26 + 11 = 37
  - 37 + 13 = 50
- **Missing Terms**: 17, 63

### 2. Sequence: ___, 4, 9, 16, 25, ___, 49, 64
- **Pattern**: This is a sequence of squares of natural numbers.
  - \(2^2 = 4\)
  - \(3^2 = 9\)
  - \(4^2 = 16\)
  - \(5^2 = 25\)
  - \(6^2 = 36\)
  - \(7^2 = 49\)
  - \(8^2 = 64\)
- **Missing Terms**: 1, 36

### 3. Sequence: 3, 12, 27, 48, ___, 108, 147, ___
- **Pattern**: Each term follows the pattern \( n^3 + 2 \times n \), where \( n \) is the position in the sequence starting from 1.
  - \(1^3 + 2 \times 1 = 3\)
  - \(2^3 + 2 \times 2 = 12\)
  - \(3^3 + 2 \times 3 = 27\)
  - \(4^3 + 2 \times 4 = 48\)
  - \(5^3 + 2 \times 5 = 75\)
  - \(6^3 + 2 \times 6 = 108\)
  - \(7^3 + 2 \times 7 = 147\)
- **Missing Terms**: 75, 196

### 4. Sequence: 1, 3, 9, ___, 81, ___, 729
- **Pattern**: Each term is \( 3^n \) where \( n \) is the position in the sequence.
  - \( 3^0 = 1 \)
  - \( 3^1 = 3 \)
  - \( 3^2 = 9 \)
  - \( 3^3 = 27 \)
  - \( 3^4 = 81 \)
  - \( 3^5 = 243 \)
  - \( 3^6 = 729 \)
- **Missing Terms**: 27, 243

### 5. Sequence: 3, 5, 11, ___, 29, 41, ___, 89
- **Pattern**: The sequence is made up of prime numbers.
  - \(3, 5, 11, 17, 29, 41, 59, 89\)
- **Missing Terms**: 17, 59

### 6. Sequence: 10, 17, 26, ___, ___, 61, 74, ___
- **Pattern**: The difference between each term is an increasing odd number.
  - \(10 + 7 = 17\)
  - \(17 + 9 = 26\)
  - \(26 + 11 = 37\)
  - \(37 + 13 = 50\)
  - \(50 + 15 = 65\)
  - \(65 + 17 = 82\)
- **Missing Terms**: 37, 50, 82

### 7. Sequence: ___, 7, 11, 19, ___, 39, ___, 79
- **Pattern**: The difference between terms is alternating between 4 and 8.
  - \(7 - 4 = 3\)
  - \(11 + 8 = 19\)
  - \(19 + 8 = 27\)
  - \(39 + 8 = 47\)
  - \(47 + 8 = 55\)
  - \(79\)
- **Missing Terms**: 3, 27, 47

### 8. Sequence: 1, 1, 2, ___, 5, 8, ___, 21, ___
- **Pattern**: This is the Fibonacci sequence where each term is the sum of the two preceding ones.
  - \(1 + 1 = 2\)
  - \(1 + 2 = 3\)
  - \(2 + 3 = 5\)
  - \(3 + 5 = 8\)
  - \(5 + 8 = 13\)
  - \(8 + 13 = 21\)
  - \(13 + 21 = 34\)
- **Missing Terms**: 3, 13, 34

Would you like more details or have any questions about these patterns? Here are 5 related questions to expand your understanding:

1. How can you identify different types of sequences such as arithmetic, geometric, and others?
2. Can you derive the nth term formula for an arithmetic sequence?
3. What is the general rule for prime number sequences?
4. How are Fibonacci sequences applied in real-world situations?
5. What are the steps to determine the common difference in a sequence with increasing gaps?

**Tip:** When faced with a sequence problem, first look for a pattern in the differences between terms. If the differences are constant, it’s arithmetic; if they multiply by a constant, it’s geometric; otherwise, consider other types of sequences like quadratic, prime, or Fibonacci.

Learn how to identify patterns and find missing terms in sequences with detailed examples. This problem covers arithmetic, geometric, prime number, and Fibonacci sequences. Suitable for Grades 7-9.

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