Let's tackle the problem step-by-step:
### Part a: Arithmetic Sequence
1. Given Information:
- The third term ([tex]\(a_3\)[/tex]) of the arithmetic sequence is 8.
- The common difference ([tex]\(d\)[/tex]) is 5.
2. Finding the First Term ([tex]\(a_1\)[/tex]):
- The general term of an arithmetic sequence is given by:
[tex]\[
a_n = a_1 + (n-1) \cdot d
\][/tex]
- For the third term ([tex]\(a_3\)[/tex]):
[tex]\[
a_3 = a_1 + 2 \cdot d
\][/tex]
- Substituting the known values:
[tex]\[
8 = a_1 + 2 \cdot 5
\][/tex]
[tex]\[
8 = a_1 + 10
\][/tex]
- Solving for [tex]\(a_1\)[/tex]:
[tex]\[
a_1 = 8 - 10 = -2
\][/tex]
3. Generating the First Five Terms:
- Now that we know [tex]\(a_1 = -2\)[/tex] and [tex]\(d = 5\)[/tex], we can find the first five terms ([tex]\(a_1, a_2, a_3, a_4, a_5\)[/tex]):
[tex]\[
a_1 = -2
\][/tex]
[tex]\[
a_2 = a_1 + d = -2 + 5 = 3
\][/tex]
[tex]\[
a_3 = a_2 + d = 3 + 5 = 8
\][/tex]
[tex]\[
a_4 = a_3 + d = 8 + 5 = 13
\][/tex]
[tex]\[
a_5 = a_4 + d = 13 + 5 = 18
\][/tex]
- Thus, the first five terms of the arithmetic sequence are:
[tex]\[
[-2, 3, 8, 13, 18]
\][/tex]
### Part b: Geometric Sequence
1. Given Information:
- The fifth term ([tex]\(g_5\)[/tex]) of the geometric sequence is [tex]\(\frac{1}{3}\)[/tex].
- The common ratio ([tex]\(r\)[/tex]) is [tex]\(\frac{1}{3}\)[/tex].
2. Finding the First Term ([tex]\(g_1\)[/tex]):
- The general term of a geometric sequence is given by:
[tex]\[
g_n = g_1 \cdot r^{n-1}
\][/tex]
- For the fifth term ([tex]\(g_5\)[/tex]):
[tex]\[
g_5 = g_1 \cdot r^4
\][/tex]
- Substituting the known values:
[tex]\[
\frac{1}{3} = g_1 \cdot \left(\frac{1}{3}\right)^4
\][/tex]
[tex]\[
\frac{1}{3} = g_1 \cdot \frac{1}{81}
\][/tex]
- Solving for [tex]\(g_1\)[/tex]:
[tex]\[
g_1 = \frac{1}{3} \cdot 81 = 27
\][/tex]
3. Generating the First Five Terms:
- Now that we know [tex]\(g_1 = 27\)[/tex] and [tex]\(r = \frac{1}{3}\)[/tex], we can find the first five terms ([tex]\(g_1, g_2, g_3, g_4, g_5\)[/tex]):
[tex]\[
g_1 = 27
\][/tex]
[tex]\[
g_2 = g_1 \cdot r = 27 \cdot \frac{1}{3} = 9
\][/tex]
[tex]\[
g_3 = g_2 \cdot r = 9 \cdot \frac{1}{3} = 3
\][/tex]
[tex]\[
g_4 = g_3 \cdot r = 3 \cdot \frac{1}{3} = 1
\][/tex]
[tex]\[
g_5 = g_4 \cdot r = 1 \cdot \frac{1}{3} = \frac{1}{3}
\][/tex]
- Thus, the first five terms of the geometric sequence are:
[tex]\[
[27, 9, 3, 1, \frac{1}{3}]
\][/tex]
### Conclusion
The sequences based on the given information are:
- Arithmetic Sequence: [tex]\([-2, 3, 8, 13, 18]\)[/tex]
- Geometric Sequence: [tex]\([27, 9, 3, 1, \frac{1}{3}]\)[/tex]
