6, 12, 18, 24, 30, ___
*

32

34

36

38

2, 7, 14, 23, 34, ___
*

47

68

782

1159

5, 6, 10, 19, 35, ___
*

38

53

60

64

1, 5, 12, 22, 35, ___
*

51

70

75

114

6, 9, 15, 24, 36, ___
*

51

53

60

69

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.

Explore how to solve arithmetic sequence problems step-by-step. This page covers identifying patterns, calculating sequence terms, and determining the next number in a sequence. Suitable for Grades 7-9 students.

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