Find the third, fourth, and fifth terms of the sequence defined by
a1 = 2,

a2 = 4,
and
an = 2an − 1 − an − 2
for
n ≥ 3.

a3 =
a4 =
a5 =

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.

Explore how to find the third, fourth, and fifth terms of a sequence defined by a recurrence relation. Step-by-step solution with calculations and patterns observed. Suitable for high school students and anyone interested in sequence analysis.

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