Sure! Let's analyze each sequence to determine whether it is geometric and, if it is, find the common ratio.
### Sequence (a) [tex]\( 4, 20, 100, 500, \ldots \)[/tex]
To determine if a sequence is geometric, we need to check if each term after the first is the result of multiplying the previous term by a constant ratio [tex]\( r \)[/tex].
1. Calculate the ratio between the first two terms:
[tex]\[
\frac{20}{4} = 5
\][/tex]
2. Calculate the ratio between the second and third terms:
[tex]\[
\frac{100}{20} = 5
\][/tex]
3. Calculate the ratio between the third and fourth terms:
[tex]\[
\frac{500}{100} = 5
\][/tex]
Since the ratio is consistent (always 5), this sequence is geometric.
- Common ratio: [tex]\( r = 5 \)[/tex]
- Conclusion: Geometric
### Sequence (b) [tex]\( 5, 10, 15, 20, \ldots \)[/tex]
Let's check the ratios between consecutive terms.
1. Calculate the ratio between the first two terms:
[tex]\[
\frac{10}{5} = 2
\][/tex]
2. Calculate the ratio between the second and third terms:
[tex]\[
\frac{15}{10} = 1.5
\][/tex]
3. Calculate the ratio between the third and fourth terms:
[tex]\[
\frac{20}{15} \approx 1.333
\][/tex]
The ratios are not consistent, hence this sequence is not geometric.
- Conclusion: Not geometric
### Sequence (c) [tex]\( 256, -64, 16, -4, \ldots \)[/tex]
Let's check the ratios between consecutive terms.
1. Calculate the ratio between the first two terms:
[tex]\[
\frac{-64}{256} = -0.25
\][/tex]
2. Calculate the ratio between the second and third terms:
[tex]\[
\frac{16}{-64} = -0.25
\][/tex]
3. Calculate the ratio between the third and fourth terms:
[tex]\[
\frac{-4}{16} = -0.25
\][/tex]
Since the ratio is consistent (always [tex]\(-0.25\)[/tex]), this sequence is geometric.
- Common ratio: [tex]\( r = -0.25 \)[/tex]
- Conclusion: Geometric
### Summary:
- Sequence (a): Geometric, [tex]\( r = 5 \)[/tex]
- Sequence (b): Not geometric
- Sequence (c): Geometric, [tex]\( r = -0.25 \)[/tex]
