Let's solve for $a_3$ in the given recursive sequence.

The information provided is:

1. Initial terms: $a_1 = -1$ and $a_2 = 4$.
2. Recursive formula: $a_{n+1} = (-1)^n a_{n-1} + \frac{n}{a_n}$

Step-by-Step Calculation of $a_3$

To find $a_3$, we substitute $n = 2$ into the recursive formula:

$a_3 = (-1)^2 a_1 + \frac{2}{a_2}$

Now, let’s plug in the known values:

1. $(-1)^2 = 1$, so $(-1)^2 a_1 = 1 \cdot (-1) = -1$.
2. Substitute $a_2 = 4$ into $\frac{2}{a_2}$: $\frac{2}{4} = 0.5$

Therefore:

$a_3 = -1 + 0.5 = -0.5$

Answer

The next term, $a_3$, is $-0.5$.

Would you like more details on the recursive calculation process or have any other questions?

Here are five follow-up questions to explore the sequence further:

1. What would be the value of $a_4$ in the sequence using the recursive formula?
2. How would the sequence behave if the initial terms were different?
3. Can you determine if this sequence converges or diverges over time?
4. How would the recursive formula change if $(-1)^n$ were replaced with $(-1)^{n+1}$?
5. What impact does the $\frac{n}{a_n}$ term have on the sequence’s behavior?

Tip: In recursive sequences, tracking each term systematically helps reveal patterns or convergence properties.