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

*a*1 = 1, *a*2 = 3,

 and 

*an* = (−1)*nan* − 1 + *an* − 2

 for 

*n* ≥ 3.

 


*a*3

 = 

*a*4

 = 

*a*5

 = 

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.

Learn how to find the third, fourth, and fifth terms of a sequence defined by a recurrence relation $a_n = (-1)^n a_{n-1} + a_{n-2}$ with initial terms $a_1 = 1$ and $a_2 = 3$. This problem covers key concepts in recurrence relations and sequences. Suitable for students in Grades 10-12.

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