Sure, let's analyze each sequence step-by-step to determine whether they are arithmetic or geometric.
### Sequence 1: -10, 20, -40, 80, ...
To determine if this sequence is geometric, we need to check if there is a common ratio (r) such that each term is obtained by multiplying the previous term by this ratio.
Let's calculate the ratio between consecutive terms:
1. The ratio between the second and first term:
[tex]\[
r_1 = \frac{20}{-10} = -2
\][/tex]
2. The ratio between the third and second term:
[tex]\[
r_2 = \frac{-40}{20} = -2
\][/tex]
3. The ratio between the fourth and third term:
[tex]\[
r_3 = \frac{80}{-40} = -2
\][/tex]
Since [tex]\( r_1 = r_2 = r_3 = -2 \)[/tex], the sequence has a common ratio of -2. Therefore, Sequence 1 is a geometric sequence.
### Sequence 2: 15, -5, -25, -45, ...
To determine if this sequence is arithmetic, we need to check if there is a common difference (d) such that each term is obtained by adding this difference to the previous term.
Let's calculate the difference between consecutive terms:
1. The difference between the second and first term:
[tex]\[
d_1 = -5 - 15 = -20
\][/tex]
2. The difference between the third and second term:
[tex]\[
d_2 = -25 - (-5) = -20
\][/tex]
3. The difference between the fourth and third term:
[tex]\[
d_3 = -45 - (-25) = -20
\][/tex]
Since [tex]\( d_1 = d_2 = d_3 = -20 \)[/tex], the sequence has a common difference of -20. Therefore, Sequence 2 is an arithmetic sequence.
### Summary
- Sequence 1: Geometric sequence with a common ratio of -2.
- Sequence 2: Arithmetic sequence with a common difference of -20.
I hope this explanation clarifies your question! If you have any further doubts, feel free to ask.
