To determine the sum of the first 6 terms of the geometric sequence $4, -4, 4, \dots$, let’s analyze and solve step-by-step.

Step 1: Identify the first term and common ratio

1. The first term $a$ is $4$.
2. The common ratio $r$ is found by dividing the second term by the first term:
   $r = \frac{-4}{4} = -1$

Step 2: Use the formula for the sum of a geometric series

The formula for the sum of the first $n$ terms of a geometric sequence is:
$S_n = a \frac{1 - r^n}{1 - r}, \quad \text{where } r \neq 1$
Here:

• $a = 4$ (first term),
• $r = -1$ (common ratio),
• $n = 6$ (number of terms).

Step 3: Substitute values into the formula

$S_6 = 4 \frac{1 - (-1)^6}{1 - (-1)}$

1. Calculate $(-1)^6$:
   $(-1)^6 = 1$
2. Substitute this into the formula:
   $S_6 = 4 \frac{1 - 1}{1 + 1}$
3. Simplify the terms:
   $S_6 = 4 \frac{0}{2} = 0$

Final Answer:

The sum of the first 6 terms of the sequence is:
$S_6 = 0$

Would you like more details about geometric sequences or the steps above?

Here are related questions for practice:

1. What is the sum of the first 8 terms of the sequence $4, -4, 4, -4, \dots$?
2. How do you find the common ratio of a geometric sequence?
3. If $a = 3$ and $r = -2$, what is $S_5$?
4. What happens to the sum of a geometric series when $r = 1$?
5. How can you determine if a sequence is geometric?

Tip: The common ratio $r$ determines whether the sequence alternates (negative $r$) or grows consistently (positive $r$).