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.