To define a geometric sequence recursively, you start with an initial term and a common ratio. The general form for the recursive definition of a geometric sequence is:
[tex]\[ f(1) = a \][/tex]
[tex]\[ f(n) = f(n-1) \cdot r \quad \text{for} \quad n > 1 \][/tex]
Where:
- [tex]\(a\)[/tex] is the first term,
- [tex]\(r\)[/tex] is the common ratio.
Given that the first term [tex]\(a\)[/tex] is [tex]\(\frac{1}{5}\)[/tex] and the common ratio [tex]\(r\)[/tex] is 5, the recursive definition for the sequence is:
[tex]\[ f(1) = \frac{1}{5} \][/tex]
[tex]\[ f(n) = f(n-1) \cdot 5 \quad \text{for} \quad n > 1 \][/tex]
Now, let's use this definition to find the first few terms of the sequence.
1. First term:
[tex]\[ f(1) = \frac{1}{5} = 0.2 \][/tex]
2. Second term:
[tex]\[ f(2) = f(1) \cdot 5 = \left(\frac{1}{5}\right) \cdot 5 = 1.0 \][/tex]
3. Third term:
[tex]\[ f(3) = f(2) \cdot 5 = 1 \cdot 5 = 5.0 \][/tex]
4. Fourth term:
[tex]\[ f(4) = f(3) \cdot 5 = 5 \cdot 5 = 25.0 \][/tex]
5. Fifth term:
[tex]\[ f(5) = f(4) \cdot 5 = 25 \cdot 5 = 125.0 \][/tex]
Therefore, the first five terms of the geometric sequence are:
[tex]\[
0.2, 1.0, 5.0, 25.0, 125.0
\][/tex]
These terms satisfy the conditions of the problem and correctly illustrate the recursive definition of the geometric sequence.
