To identify the first term ([tex]\(a_1\)[/tex]) and the common ratio ([tex]\(r\)[/tex]) for each given geometric sequence, we follow this method.
### Sequence 1: [tex]\(1, 3, 9, 27, \ldots\)[/tex]
1. First Term ([tex]\(a_1\)[/tex]): The first term of the sequence is the initial term given, which is 1.
2. Common Ratio ([tex]\(r\)[/tex]): The common ratio is found by dividing the second term by the first term.
[tex]\[
r = \frac{3}{1} = 3
\][/tex]
Thus, for the first sequence:
[tex]\[
a_1 = 1, \quad r = 3.0
\][/tex]
### Sequence 2: [tex]\(8, 4, 2, 1, \ldots\)[/tex]
1. First Term ([tex]\(a_1\)[/tex]): The first term of the sequence is the initial term given, which is 8.
2. Common Ratio ([tex]\(r\)[/tex]): The common ratio is found by dividing the second term by the first term.
[tex]\[
r = \frac{4}{8} = 0.5
\][/tex]
Thus, for the second sequence:
[tex]\[
a_1 = 8, \quad r = 0.5
\][/tex]
### Sequence 3: [tex]\(4, -16, 64, -256, \ldots\)[/tex]
1. First Term ([tex]\(a_1\)[/tex]): The first term of the sequence is the initial term given, which is 4.
2. Common Ratio ([tex]\(r\)[/tex]): The common ratio is found by dividing the second term by the first term.
[tex]\[
r = \frac{-16}{4} = -4
\][/tex]
Thus, for the third sequence:
[tex]\[
a_1 = 4, \quad r = -4.0
\][/tex]
Summarizing all the sequences, we get:
- For the sequence [tex]\(1, 3, 9, 27, \ldots\)[/tex]:
[tex]\[
a_1 = 1, \quad r = 3.0
\][/tex]
- For the sequence [tex]\(8, 4, 2, 1, \ldots\)[/tex]:
[tex]\[
a_1 = 8, \quad r = 0.5
\][/tex]
- For the sequence [tex]\(4, -16, 64, -256, \ldots\)[/tex]:
[tex]\[
a_1 = 4, \quad r = -4.0
\][/tex]
