In the image, it appears that you are working with sequences, both arithmetic and geometric. Let's go over the problems shown in the image:

2. Geometric Sequence

• First term: -9
• Common ratio: -2

This is a geometric sequence because each term is multiplied by a common ratio of -2 to get the next term.

• The terms are: $-9, 18, -36, 72, -144, \dots$ Each term is obtained by multiplying the previous term by -2.

3. Arithmetic Sequence

• First term: 0
• Common difference: -6

This is an arithmetic sequence because each term is found by subtracting 6 from the previous term.

• The terms are: $0, -6, -12, -18, -24, \dots$ Each term is found by adding the common difference of -6 to the previous term.

4. Geometric Sequence

• First term: -3
• Common ratio: -$\frac{1}{4}$

This is another geometric sequence where each term is multiplied by -$\frac{1}{4}$.

• The terms will be: $-3, \frac{3}{4}, -\frac{3}{16}, \dots$ Each term is obtained by multiplying the previous term by -$\frac{1}{4}$.

Let me know if you'd like further explanations on any of these sequences or have any specific questions!

5 Questions for Exploration:

1. How do you derive the formula for the nth term in a geometric sequence?
2. What is the sum of the first n terms of an arithmetic sequence?
3. How can you determine if a sequence is geometric or arithmetic just by looking at a few terms?
4. How does the common ratio affect the behavior of a geometric sequence over time?
5. Can an arithmetic sequence ever have a common difference of 0? What would that mean?

Tip:

When dealing with geometric sequences, if the common ratio's absolute value is less than 1, the terms will get smaller and smaller, eventually approaching zero.