Here are the solutions to the sequences provided in the image:

10. Sequence: 1, 4, 10, 22, 46, ?

• Explanation: The differences between consecutive terms are:
  • $4 - 1 = 3$,
  • $10 - 4 = 6$,
  • $22 - 10 = 12$,
  • $46 - 22 = 24$.
  • The differences are doubling: $3, 6, 12, 24$.
  • Next difference: $24 \times 2 = 48$.
  • Next term: $46 + 48 = 94$.

Answer: $\mathbf{94}$ (Option B).

11. Sequence: 4, 9, 25, ?, 121, 169, 289, 361

• Explanation: The terms are squares of prime numbers:
  • $4 = 2^2$,
  • $9 = 3^2$,
  • $25 = 5^2$,
  • $? = 7^2 = 49$,
  • $121 = 11^2$,
  • $169 = 13^2$, etc.

Answer: $\mathbf{49}$ (Option A).

12. Sequence: 1, 1, 2, 6, 24, ?, 720

• Explanation: The terms represent factorials:
  • $1 = 0!$,
  • $1 = 1!$,
  • $2 = 2!$,
  • $6 = 3!$,
  • $24 = 4!$,
  • $? = 5! = 120$,
  • $720 = 6!$.

Answer: $\mathbf{120}$.

13. Sequence: 4, 6, 9, 13 $\frac{1}{2}$, ?

• Explanation:
  • Differences: $6 - 4 = 2$, $9 - 6 = 3$, $13 \frac{1}{2} - 9 = 4 \frac{1}{2}$.
  • Next difference: $4 \frac{1}{2} + 1 = 5 \frac{1}{2}$.
  • Next term: $13 \frac{1}{2} + 5 \frac{1}{2} = 19$.

Answer: $\mathbf{19}$ (Option B).

Do you have any questions or want detailed steps for any of these?

Related Questions:

1. How do you recognize patterns in number sequences?
2. Can these methods apply to sequences with more complex rules?
3. What are factorials and how are they used in sequences?
4. How do prime numbers contribute to mathematical patterns?
5. Are there sequences with mixed patterns, and how are they solved?

Tip:

Always start by analyzing the differences between terms to spot arithmetic or geometric patterns. If differences vary, check for factorials, squares, cubes, or primes.