To find the 6th term of the sequence $3, -6, 12, -24, \dots$, we can recognize this as a geometric sequence.

Step 1: Identify the first term and common ratio

• First term ($a_1$): $3$
• Common ratio ($r$): To find $r$, divide the second term by the first term: $r = \frac{-6}{3} = -2$

Step 2: Find the 6th term

The general formula for the $n$th term of a geometric sequence is: $a_n = a_1 \times r^{n-1}$ For the 6th term ($a_6$): $a_6 = 3 \times (-2)^{6-1} = 3 \times (-2)^5$ $a_6 = 3 \times (-32) = -96$

Answer:

The 6th term of the sequence is -96.

Would you like further details or have any questions? Here are some related questions you might find interesting:

1. How does the sign of the common ratio affect the sequence?
2. What is the sum of the first 6 terms of this geometric sequence?
3. How would you find the $n$th term if the sequence alternated differently?
4. What happens to the sequence if the common ratio is fractional?
5. Can a geometric sequence have a common ratio that is a complex number?

Tip: Pay close attention to the sign of the common ratio, as it determines whether the sequence alternates between positive and negative terms.