find the indicated term of the geometric sequence a4=1/2, r=-4, n=9

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

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

Where:
- \( a_n \) is the \( n \)th term of the sequence,
- \( a_1 \) is the first term,
- \( r \) is the common ratio,
- \( n \) is the term number.

Given:
- \( a_4 = \frac{1}{2} \)
- \( r = -4 \)
- \( n = 9 \)

First, we need to find the first term \( a_1 \) using the information provided for \( a_4 \):

\[
a_4 = a_1 \cdot r^{3}
\]

Substitute the known values:

\[
\frac{1}{2} = a_1 \cdot (-4)^3
\]

Since \( (-4)^3 = -64 \), we have:

\[
\frac{1}{2} = a_1 \cdot (-64)
\]

To find \( a_1 \), solve for \( a_1 \):

\[
a_1 = \frac{\frac{1}{2}}{-64} = -\frac{1}{128}
\]

Now that we have \( a_1 = -\frac{1}{128} \), we can find the 9th term \( a_9 \) using the formula:

\[
a_9 = a_1 \cdot r^{8}
\]

Substituting the values:

\[
a_9 = -\frac{1}{128} \cdot (-4)^{8}
\]

Since \( (-4)^8 = 65536 \), we get:

\[
a_9 = -\frac{1}{128} \cdot 65536
\]

\[
a_9 = -512
\]

So, the 9th term \( a_9 \) is \(-512\).

Would you like any further explanation or details on this? 

Here are five related questions you might find interesting:
1. How do you find the sum of a finite geometric series?
2. What is the difference between an arithmetic and a geometric sequence?
3. How does the common ratio affect the behavior of a geometric sequence?
4. Can the common ratio of a geometric sequence be zero? What happens if it is?
5. How would you find the nth term if the common ratio is a fraction?

**Tip:** In geometric sequences, if the common ratio's absolute value is greater than 1, the terms will grow exponentially; if it is between 0 and 1, the terms will decay exponentially.

Learn how to find the 9th term of a geometric sequence where the 4th term is 1/2 and the common ratio is -4. Detailed step-by-step solution provided.

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