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.