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.