Let's analyze each sequence, derive the recursive formula, and determine the next two terms.

1. Sequence: 5, 9, 13, 17, ...

Recursive Formula:
This sequence is arithmetic, where each term increases by a common difference of 4.

$a_n = a_{n-1} + 4$

Next Two Terms:

• $a_5 = 17 + 4 = 21$
• $a_6 = 21 + 4 = 25$

So, the next two terms are 21 and 25.

2. Sequence: 8, 14, 26, 50, ...

Recursive Formula:
This sequence is neither arithmetic nor geometric but appears to involve a pattern where the difference between consecutive terms increases. Let's calculate the differences between terms:

• $14 - 8 = 6$
• $26 - 14 = 12$
• $50 - 26 = 24$

The differences (6, 12, 24) seem to be doubling.

$a_n = a_{n-1} + 2^{n+1}$

Next Two Terms:

• $a_5 = 50 + 48 = 98$
• $a_6 = 98 + 96 = 194$

So, the next two terms are 98 and 194.

3. Sequence: 7, 1, -5, -11, ...

Recursive Formula:
This sequence is arithmetic, with each term decreasing by a constant difference of 6.

$a_n = a_{n-1} - 6$

Next Two Terms:

• $a_5 = -11 - 6 = -17$
• $a_6 = -17 - 6 = -23$

So, the next two terms are -17 and -23.

4. Sequence: 3, 10, 24, 52, ...

Recursive Formula:
The differences between consecutive terms are:

• $10 - 3 = 7$
• $24 - 10 = 14$
• $52 - 24 = 28$

The differences (7, 14, 28) double each time, indicating a pattern similar to the second sequence.

$a_n = a_{n-1} + 2^n \times 7$

Next Two Terms:

• $a_5 = 52 + 56 = 108$
• $a_6 = 108 + 112 = 220$

So, the next two terms are 108 and 220.

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Do you want details or have any questions?

Here are 8 related questions to explore further:

1. How can the general formula for an arithmetic sequence be derived?
2. What is the formula for the nth term of an arithmetic sequence?
3. How does the recursive formula differ from the explicit formula?
4. What are the properties of geometric sequences?
5. How can we identify if a sequence is arithmetic or geometric?
6. What role does the common difference play in an arithmetic sequence?
7. How do you determine the next term in a sequence using a recursive formula?
8. What are the applications of sequences in real-life problems?

Tip: For non-arithmetic sequences, always check if the differences or ratios between terms follow a consistent pattern to identify the recursive formula.