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The problem asks to find the common ratio of a geometric progression, given that it is negative, where the first term is 8, and the fifth term is \(\frac{1}{2}\).

### Solution:

In a geometric progression (GP), the \(n\)th term is given by:
\[
a_n = a_1 \cdot r^{n-1}
\]
where:
- \(a_1\) is the first term,
- \(r\) is the common ratio, and
- \(n\) is the term number.

Given:
- \(a_1 = 8\)
- \(a_5 = \frac{1}{2}\)
- \(n = 5\)

Substituting the given values into the formula for the fifth term:
\[
\frac{1}{2} = 8 \cdot r^{5-1}
\]
\[
\frac{1}{2} = 8 \cdot r^4
\]
Now, solve for \(r^4\):
\[
r^4 = \frac{1}{2} \times \frac{1}{8} = \frac{1}{16}
\]
Taking the fourth root of both sides:
\[
r = \sqrt[4]{\frac{1}{16}} = \sqrt[4]{\left(\frac{1}{2}\right)^4} = \frac{1}{2}
\]

Since the problem specifies that the common ratio is negative:
\[
r = -\frac{1}{2}
\]

Thus, the common ratio \(r\) is \(-\frac{1}{2}\).

Would you like more details or have any questions?

Here are five related questions:

1. How do you determine the nth term of a geometric progression if the common ratio is known?
2. What happens to a geometric progression if the common ratio is positive but less than 1?
3. Can a geometric progression have a zero common ratio? Why or why not?
4. How would you solve for the common ratio if the 10th term of the sequence was provided instead of the 5th?
5. How does the sign of the common ratio affect the progression of terms in the sequence?

**Tip:** When solving problems involving geometric progressions, always check whether the common ratio is positive or negative, as this will affect the sequence's pattern.

Find the common ratio, given that it is negative, of a geometric progression whose first term is 8 and whose fifth term is 1/2.

This problem involves finding the negative common ratio of a geometric progression where the first term is 8 and the fifth term is 1/2. The solution includes the application of the formula for the nth term of a geometric sequence.

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