Let's solve each sequence one by one to identify the pattern and determine the correct next number:

1. Sequence: 6, 12, 18, 24, 30, ___

   • This is an arithmetic sequence with a common difference of 6.
   • Next term: $30 + 6 = 36$
   • Answer: 36

2. Sequence: 2, 7, 14, 23, 34, ___

   • The pattern is increasing by consecutive odd numbers: $+5, +7, +9, +11, +13$
   • Next term: $34 + 13 = 47$
   • Answer: 47

3. Sequence: 5, 6, 10, 19, 35, ___

   • The pattern involves adding consecutive terms and increasing increments: $+1, +4, +9, +16$
   • Next term: $35 + 18 = 53$ (increment by 18, continuing the pattern of squares: $1^2, 2^2, 3^2, 4^2, 5^2$)
   • Answer: 53

4. Sequence: 1, 5, 12, 22, 35, ___

   • The pattern is the sum of increasing differences: $+4, +7, +10, +13$
   • Next term: $35 + 16 = 51$ (increasing by $+3$ each time)
   • Answer: 51

5. Sequence: 6, 9, 15, 24, 36, ___

   • The pattern involves adding consecutive terms and an increasing number: $+3, +6, +9, +12, +15$
   • Next term: $36 + 15 = 51$
   • Answer: 51

Would you like more details or have any other questions?

Here are 5 related questions:

1. What are the common methods to identify patterns in sequences?
2. How do arithmetic sequences differ from geometric sequences?
3. Can a sequence have more than one correct next term? Why or why not?
4. How can you recognize if a sequence follows a quadratic pattern?
5. What is the difference between a linear sequence and a non-linear sequence?

Tip: When working with sequences, try looking at the differences between terms to uncover any hidden patterns. This approach can help you determine the rule governing the sequence.