We are given a geometric sequence where $a_1 = 2$ and the common ratio $r = -6$. We need to find the sum of the first five terms.

The formula for the sum $S_n$ of the first $n$ terms of a geometric sequence is:

$S_n = \frac{a_1(1 - r^n)}{1 - r} \quad \text{(for } r \neq 1 \text{)}$

For this problem:

• $a_1 = 2$
• $r = -6$
• $n = 5$

Substituting these values into the formula:

$S_5 = \frac{2(1 - (-6)^5)}{1 - (-6)}$

Now, I'll calculate the exact sum.The sum of the first five terms of the geometric sequence is $2,222$.

Would you like further details on this solution, or do you have any other questions?

Here are 5 related questions to consider:

1. How would the sum change if the common ratio $r$ were positive?
2. What is the formula for the sum if $n$ is not given and we have to compute indefinitely many terms?
3. How does changing $a_1$ affect the sum of the sequence?
4. Can you explain the difference between arithmetic and geometric sequences?
5. What would happen if the common ratio $r = 1$?

Tip: When working with geometric sequences, if the absolute value of the ratio $|r|$ is greater than 1, the terms grow (or shrink) rapidly.