To find the 6th term of the geometric sequence, we follow these steps:

Step 1: Identify the first term ($a$) and common ratio ($r$):

The sequence is: $-\frac{4}{243}, \frac{4}{81}, -\frac{4}{27}, \dots$

1. The first term ($a$) is: $a = -\frac{4}{243}$

2. To find the common ratio ($r$), divide the second term by the first term: $r = \frac{\frac{4}{81}}{-\frac{4}{243}} = \frac{4}{81} \cdot \frac{-243}{4} = -3$

Step 2: Use the formula for the $n$-th term of a geometric sequence:

The formula is: $a_n = a \cdot r^{n-1}$

Here, $n = 6$. Substituting the values of $a = -\frac{4}{243}$, $r = -3$, and $n = 6$: $a_6 = -\frac{4}{243} \cdot (-3)^{6-1}$

Step 3: Simplify the exponent:

$(-3)^{6-1} = (-3)^5 = -243$

Step 4: Calculate $a_6$:

$a_6 = -\frac{4}{243} \cdot -243$ $a_6 = \frac{972}{243}$ $a_6 = 4$

Final Answer:

The 6th term of the sequence is: $\boxed{4}$

Would you like further clarification or details on this solution?

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Related Questions:

1. How is the common ratio in a geometric sequence determined?
2. What are the properties of a geometric sequence with alternating signs?
3. Can you derive a general formula for the sum of the first $n$ terms of this sequence?
4. How would the sequence change if the common ratio were positive?
5. What are the applications of geometric sequences in real-world scenarios?

Tip:

Always double-check the common ratio calculation when solving geometric sequences to avoid sign errors!