To find the third, fourth, and fifth terms of the sequence defined by the recurrence relation $a_n = 2a_{n-1} - a_{n-2}$ for $n \geq 3$, we can calculate each term step by step:

Given: $a_1 = 2, \quad a_2 = 4$

Finding $a_3$:

$a_3 = 2a_2 - a_1$ Substituting the values: $a_3 = 2(4) - 2 = 8 - 2 = 6$

Finding $a_4$:

$a_4 = 2a_3 - a_2$ Substituting the values: $a_4 = 2(6) - 4 = 12 - 4 = 8$

Finding $a_5$:

$a_5 = 2a_4 - a_3$ Substituting the values: $a_5 = 2(8) - 6 = 16 - 6 = 10$

So, the third, fourth, and fifth terms of the sequence are: $a_3 = 6, \quad a_4 = 8, \quad a_5 = 10$

Would you like more details or have any questions? Here are some related questions:

1. How can you generalize the formula for the $n$th term of the sequence?
2. What pattern do you observe in the sequence as it progresses?
3. Can you determine $a_6$ using the same recurrence relation?
4. How does the sequence change if the initial terms $a_1$ and $a_2$ are altered?
5. What kind of sequences use recurrence relations similar to this one?

Tip: When dealing with sequences defined by recurrence relations, it can be useful to calculate the first few terms by hand to identify patterns or to validate a conjecture about the sequence's behavior.