To find the first four terms of the sequence defined by the given recursive formula:

$a_1 = 2 \quad \text{and} \quad a_n = a_{n-1} + 4 \quad \text{for } n \geq 2,$

we proceed as follows:

Step-by-step solution:

1. First term: $a_1 = 2$

2. Second term: $a_2 = a_1 + 4 = 2 + 4 = 6$

3. Third term: $a_3 = a_2 + 4 = 6 + 4 = 10$

4. Fourth term: $a_4 = a_3 + 4 = 10 + 4 = 14$

Final Answer:

The first four terms of the sequence are: $2, 6, 10, 14$

Would you like to see a graph of the sequence or details about the recursive growth? Here are five related questions to expand this concept:

1. How would you generalize this sequence to find any term $a_n$ without recursion?
2. What type of sequence is this, and how can you prove it?
3. Can you calculate the sum of the first $n$ terms of this sequence?
4. How does the sequence change if the increment is modified (e.g., to $+3$ instead of $+4$)?
5. How does this sequence relate to an arithmetic progression in general?

Tip: Always verify recursion formulas by calculating multiple terms to ensure consistency.