To find the third, fourth, and fifth terms of the sequence defined by the recurrence relation

$a_n = (-1)^n a_{n-1} + a_{n-2} \quad \text{for } n \geq 3$

and given the initial terms:

$a_1 = 1, \quad a_2 = 3$

we calculate each term step by step.

Finding $a_3$:

$a_3 = (-1)^3 a_2 + a_1$ Substituting the values: $a_3 = (-1)(3) + 1 = -3 + 1 = -2$

Finding $a_4$:

$a_4 = (-1)^4 a_3 + a_2$ Substituting the values: $a_4 = (1)(-2) + 3 = -2 + 3 = 1$

Finding $a_5$:

$a_5 = (-1)^5 a_4 + a_3$ Substituting the values: $a_5 = (-1)(1) + (-2) = -1 - 2 = -3$

So, the third, fourth, and fifth terms of the sequence are:

$a_3 = -2, \quad a_4 = 1, \quad a_5 = -3$

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

1. How does the sign change due to $(-1)^n$ affect the sequence?
2. What pattern do you observe in the sequence?
3. Can you predict the behavior of $a_6$ based on the current pattern?
4. How would the sequence change if the initial terms $a_1$ and $a_2$ were different?
5. Can you find a closed-form expression for $a_n$?

Tip: When working with sequences that involve alternating signs, pay close attention to the power of $(-1)$ as it affects the result significantly depending on whether $n$ is odd or even.