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Find the missing term in each of the following sequence

Let's analyze and solve the missing terms in each sequence:

1. **Sequence: 2, 5, 10, 17, ___, 37, ___, 65, ___, ___, 145**
   - Pattern: \(n^2 + 1\) where \(n\) starts from 1 and increases by 1.
     - \(1^2 + 1 = 2\)
     - \(2^2 + 1 = 5\)
     - \(3^2 + 1 = 10\)
     - \(4^2 + 1 = 17\)
     - \(5^2 + 1 = 26\)
     - \(6^2 + 1 = 37\)
     - \(7^2 + 1 = 50\)
     - \(8^2 + 1 = 65\)
     - \(9^2 + 1 = 82\)
     - \(10^2 + 1 = 101\)
     - \(11^2 + 1 = 122\)
     - \(12^2 + 1 = 145\)
   - **Missing Terms: 26, 50, 82, 101, 122**

2. **Sequence: ___, 8, ___, 64, ___, 216, 343**
   - Pattern: These are cubes of consecutive integers.
     - \(1^3 = 1\)
     - \(2^3 = 8\)
     - \(3^3 = 27\)
     - \(4^3 = 64\)
     - \(5^3 = 125\)
     - \(6^3 = 216\)
     - \(7^3 = 343\)
   - **Missing Terms: 1, 27, 125**

3. **Sequence: 1, 1, 3, 5, 9, 17, 31, ___**
   - Pattern: Each term is the sum of the two previous terms.
     - \(1 + 1 = 2\)
     - \(1 + 2 = 3\)
     - \(2 + 3 = 5\)
     - \(3 + 5 = 8\)
     - \(5 + 8 = 13\)
     - \(8 + 13 = 21\)
     - \(13 + 21 = 34\)
   - **Missing Term: 57**

4. **Sequence: 30, 22, 14, ___, -10, -18, ___**
   - Pattern: Decreasing by 8.
     - \(30 - 8 = 22\)
     - \(22 - 8 = 14\)
     - \(14 - 8 = 6\)
     - \(-10 - 8 = -18\)
   - **Missing Terms: 6, -26**

5. **Sequence: 3, 7, 15, 31, 63, ___**
   - Pattern: \(2n + 1\)
     - \(3 \times 2 + 1 = 7\)
     - \(7 \times 2 + 1 = 15\)
     - \(15 \times 2 + 1 = 31\)
     - \(31 \times 2 + 1 = 63\)
     - \(63 \times 2 + 1 = 127\)
   - **Missing Term: 127**

6. **Sequence: 8, 5, 4, 9, 1, 7, 3, 11, ___**
   - Pattern: Alternating prime numbers starting from 2 and 3.
     - Next is 6.
   - **Missing Term: 6**

7. **Sequence: 1, 10, 11, 21, 31, 41, 1401, 16, 512**
   - Pattern: The pattern isn’t clear, but it could involve alternating primes, squares, or other operations.
   - **Answer requires further investigation.**

8. **Sequence: F, T, F, T, T, ___, F, F, F, ___, S**
   - Pattern: Alternating True (T) and False (F), and then a sequence "F, F, F".
     - **Missing Term: T, S**

9. **Sequence: J, F, M, A, M, ___, J, A, ___**
   - Pattern: First letters of months.
   - **Missing Terms: J (June), S (September)**

10. **Sequence: O, T, T, F, F, S, S, ___**
    - Pattern: First letters of number words.
      - One, Two, Three, Four, Five, Six, Seven.
    - **Missing Term: E (Eight)**

Would you like further details or explanations on any of these sequences?

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**Relative Questions:**
1. How can sequences with alternating patterns be identified?
2. What methods can be used to find missing terms in arithmetic sequences?
3. How can one determine if a sequence follows a geometric pattern?
4. What are some common sequences in mathematics and their patterns?
5. How does the Fibonacci sequence differ from other number sequences?

**Tip:** When solving sequence problems, identify if the sequence follows a specific mathematical pattern, such as arithmetic, geometric, or a combination of operations.

Explore solutions to find the missing terms in various sequences through mathematical pattern analysis. Learn how to identify arithmetic, geometric, and complex patterns in sequences. Suitable for high school students and math enthusiasts.

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