what is the 6th term of the geometric sequence where a_1=1,024 and a_4=-16

To find the 6th term of a geometric sequence, we need to first determine the common ratio \( r \) using the information provided:

1. The first term of the sequence is \( a_1 = 1024 \).
2. The fourth term of the sequence is \( a_4 = -16 \).

### Formula for the \( n \)-th Term of a Geometric Sequence

The formula for the \( n \)-th term of a geometric sequence is given by:

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

### Finding the Common Ratio \( r \)

Given \( a_4 = -16 \), we can use the formula for the \( n \)-th term:

\[
a_4 = a_1 \cdot r^{4-1} = a_1 \cdot r^3
\]

Substituting the known values:

\[
-16 = 1024 \cdot r^3
\]

To find \( r^3 \), we divide both sides by 1024:

\[
r^3 = \frac{-16}{1024} = -\frac{1}{64}
\]

Now, we solve for \( r \) by taking the cube root of both sides:

\[
r = \sqrt[3]{-\frac{1}{64}} = -\frac{1}{4}
\]

### Calculating the 6th Term \( a_6 \)

Now that we know the common ratio \( r = -\frac{1}{4} \), we can find the 6th term \( a_6 \):

\[
a_6 = a_1 \cdot r^{6-1} = 1024 \cdot \left(-\frac{1}{4}\right)^5
\]

Calculating \( \left(-\frac{1}{4}\right)^5 \):

\[
\left(-\frac{1}{4}\right)^5 = -\frac{1}{1024}
\]

Now, substitute this back into the equation for \( a_6 \):

\[
a_6 = 1024 \cdot \left(-\frac{1}{1024}\right) = -1
\]

### Conclusion

Thus, the 6th term of the geometric sequence is \(-1\).

If you have any questions or need further details, feel free to ask! Here are some questions you might consider:

1. How do you find the common ratio in a geometric sequence if given different terms?
2. What is the difference between an arithmetic sequence and a geometric sequence?
3. How can I determine the sum of a geometric sequence?
4. How do geometric sequences apply to real-world scenarios?
5. What happens to the terms of a geometric sequence if the common ratio is negative?

**Tip:** In geometric sequences, a negative common ratio results in terms alternating in sign.

Learn how to find the 6th term of a geometric sequence where the first term \( a_1 = 1024 \) and the fourth term \( a_4 = -16 \). Understand the formula for the nth term of a geometric sequence and how to determine the common ratio. Suitable for students in Grades 9-12.

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