It seems like you're working with a recurrence relation. Based on the given equations, it appears that:

1. $a_1 = 5$
2. The recurrence relation: $a_n = -6a_{n-1}$, which defines each term based on the previous one.

Let's use this to calculate $a_2$ and $a_3$:

Step 1: Calculate $a_2$

Using the recurrence relation $a_n = -6a_{n-1}$, for $n = 2$, we substitute $a_1 = 5$ into the formula:

$a_2 = -6a_1 = -6 \times 5 = -30$

Step 2: Calculate $a_3$

Now, for $n = 3$, we substitute $a_2 = -30$ into the formula:

$a_3 = -6a_2 = -6 \times (-30) = 180$

Thus:

• $a_2 = -30$
• $a_3 = 180$

Let me know if you'd like further details or explanations! Here are some related questions to explore:

1. What is the general formula for $a_n$ based on the recurrence relation?
2. How does the recurrence relation behave for higher terms like $a_4$?
3. What happens if the recurrence relation changes to $a_n = -3a_{n-1}$?
4. Can you find the sum of the first $n$ terms of this recurrence?
5. How would the series behave if the initial term $a_1$ was different, say $a_1 = 2$?

Tip: In recurrence relations, finding a general formula for $a_n$ is key to quickly calculating later terms without recursion.