Find the 12th term of the geometric sequence shown below.
minus, 9, x, to the power 8 , comma, 36, x, to the power 12 , comma, minus, 144, x, to the power 16 , comma, point, point, .
−9x 
8
 ,36x 
12
 ,−144x 
16
 ,...

The given sequence is:

\[
-9x^8, \quad 36x^{12}, \quad -144x^{16}, \quad \dots
\]

This is a geometric sequence where each term is obtained by multiplying the previous term by a constant ratio \( r \). Let's first find the common ratio \( r \) by dividing the second term by the first term:

\[
r = \frac{36x^{12}}{-9x^8} = \frac{36}{-9} \cdot \frac{x^{12}}{x^8} = -4x^4
\]

Thus, the common ratio \( r = -4x^4 \).

To find the 12th term of the sequence, we use the formula for the \(n\)-th term of a geometric sequence:

\[
a_n = a_1 \cdot r^{n-1}
\]

Here:
- \( a_1 = -9x^8 \)
- \( r = -4x^4 \)
- \( n = 12 \)

Now, we calculate the 12th term:

\[
a_{12} = -9x^8 \cdot (-4x^4)^{11}
\]

Let's break this calculation down:

1. Calculate the power of \( r \):
\[
(-4x^4)^{11} = (-4)^{11} \cdot (x^4)^{11} = -4194304 \cdot x^{44}
\]

2. Now multiply by \( a_1 \):
\[
a_{12} = -9x^8 \cdot (-4194304x^{44}) = 9 \cdot 4194304 \cdot x^{8+44} = 37748736x^{52}
\]

Thus, the 12th term of the sequence is:

\[
a_{12} = 37748736x^{52}
\]

Would you like further details or have any questions about this solution?

Here are five related questions to explore:

1. What is the common ratio of a geometric sequence?
2. How do you find the general term in a geometric sequence?
3. What happens to the terms of a geometric sequence if the common ratio is negative?
4. How can geometric sequences be applied in real-life problems?
5. What is the sum of the first \(n\) terms in a geometric sequence?

**Tip:** In a geometric sequence, the common ratio can be found by dividing any term by its preceding term.

Find the 12th term of the geometric sequence: −9x^8, 36x^12, −144x^16,...

This solution demonstrates how to find the 12th term of a geometric sequence where terms involve variables with exponents. Using the geometric sequence formula and exponent rules, the 12th term is calculated step by step. Suitable for high school students in Grades 10-12.

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