To solve this problem, let’s break down the geometric sequence given and find the required formula and term.

1. Identify the common ratio and first term

The given sequence is: $5, -10, 20, -40, \dots$

• The first term $a_1$ is 5.
• To find the common ratio $r$, we can divide the second term by the first term: $r = \frac{-10}{5} = -2$

So, the common ratio $r$ is -2.

2. Write the formula for the $n$-th term of a geometric sequence

The formula for the $n$-th term $a_n$ of a geometric sequence is: $a_n = a_1 \cdot r^{n-1}$ Substituting $a_1 = 5$ and $r = -2$: $a_n = 5 \cdot (-2)^{n-1}$

3. Find $a_9$

To find $a_9$, substitute $n = 9$ into the formula: $a_9 = 5 \cdot (-2)^{9-1}$ $a_9 = 5 \cdot (-2)^8$ Since $(-2)^8 = 256$: $a_9 = 5 \cdot 256 = 1280$

Final Answer

• The formula for the $n$-th term of the sequence is: $a_n = 5 \cdot (-2)^{n-1}$
• The ninth term $a_9$ is: $a_9 = 1280$

Would you like further clarification on any steps or details?

Here are five additional related questions:

1. How would the formula change if the common ratio were positive instead of negative?
2. Can you derive the sum of the first $n$ terms of this geometric sequence?
3. What happens to the $n$-th term as $n$ approaches infinity in a geometric sequence with $|r| < 1$?
4. How does changing the first term $a_1$ affect the entire sequence?
5. How can you determine if a sequence is geometric just by looking at the terms?

Tip: In geometric sequences, the common ratio is found by dividing any term by the previous term. This ratio remains constant throughout the sequence.