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This problem involves a geometric sequence where the fifth term is 80 and the eighth term is -640. We are asked to find the common ratio \( r \), the first term \( a_1 \), and the ninth term \( a_9 \).

### Step 1: Set up the formula for the nth term of a geometric sequence.
The nth term of a geometric sequence is given by:

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

### Step 2: Write the equations for the 5th and 8th terms.

Given that:
- The 5th term \( a_5 = 80 \)
- The 8th term \( a_8 = -640 \)

We can set up two equations using the general formula for the nth term:

\[
a_5 = a_1 \cdot r^4 = 80
\]
\[
a_8 = a_1 \cdot r^7 = -640
\]

### Step 3: Solve for the ratio \( r \).

Divide the second equation by the first equation to eliminate \( a_1 \):

\[
\frac{a_8}{a_5} = \frac{a_1 \cdot r^7}{a_1 \cdot r^4} = \frac{-640}{80}
\]

Simplifying this:

\[
r^3 = \frac{-640}{80} = -8
\]

Thus, the cube root of -8 gives us:

\[
r = -2
\]

### Step 4: Solve for \( a_1 \).

Now, substitute \( r = -2 \) into one of the original equations, say the equation for \( a_5 \):

\[
a_1 \cdot (-2)^4 = 80
\]

Simplify:

\[
a_1 \cdot 16 = 80
\]

Solving for \( a_1 \):

\[
a_1 = \frac{80}{16} = 5
\]

### Step 5: Find \( a_9 \).

Using the formula for the nth term again, we can find \( a_9 \):

\[
a_9 = a_1 \cdot r^8 = 5 \cdot (-2)^8
\]

Simplifying:

\[
a_9 = 5 \cdot 256 = 1280
\]

### Final Answers:
- \( r = -2 \)
- \( a_1 = 5 \)
- \( a_9 = 1280 \)

Would you like further details on any of the steps? Here are some related questions:

1. How do you determine the nth term in a geometric sequence?
2. What happens to a sequence when the ratio is negative?
3. How can you find the sum of the first n terms in a geometric series?
4. Can a geometric sequence have a ratio of 1 or 0?
5. What are the key differences between arithmetic and geometric sequences?

**Tip:** When solving problems involving sequences, using ratios between terms can often simplify finding the common ratio.

In a geometric sequence, the fifth term is 80 and the eighth term is -640. Find the common ratio r, the first term a1, and the ninth term a9.

Solve a geometric sequence problem where the fifth term is 80 and the eighth term is -640. Learn how to find the common ratio, the first term, and the ninth term with a step-by-step solution. Ideal for students in Grades 9-12.

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