To determine the correct recursive function that defines the given geometric sequence, we need to examine the sequence's terms and find a pattern in their growth.
Given:
- [tex]\( f(1) = 4 \)[/tex]
- [tex]\( f(2) = 20 \)[/tex]
- [tex]\( f(3) = 100 \)[/tex]
Let's analyze the given options one by one and check whether they match the provided sequence.
### Option A:
#### [tex]\( f(1) = 1 \)[/tex]
#### [tex]\( f(n) = 10 \cdot f(n-1) \)[/tex] for [tex]\( n \geq 2 \)[/tex]
For [tex]\( n = 1 \)[/tex]:
- [tex]\( f(1) = 1 \)[/tex], does not match [tex]\( f(1) = 4 \)[/tex], so this option is incorrect.
### Option B:
#### [tex]\( f(1) = 4 \)[/tex]
#### [tex]\( f(n) = 16 \cdot f(n-1) \)[/tex] for [tex]\( n \geq 2 \)[/tex]
For [tex]\( n = 2 \)[/tex]:
- [tex]\( f(2) = 16 \cdot f(1) = 16 \cdot 4 = 64 \)[/tex], does not match [tex]\( f(2) = 20 \)[/tex], so this option is incorrect.
### Option C:
#### [tex]\( f(1) = 4 \)[/tex]
#### [tex]\( f(n) = 5 \cdot f(n-1) \)[/tex] for [tex]\( n \geq 2 \)[/tex]
For [tex]\( n = 2 \)[/tex]:
- [tex]\( f(2) = 5 \cdot f(1) = 5 \cdot 4 = 20 \)[/tex], matches [tex]\( f(2) = 20 \)[/tex].
For [tex]\( n = 3 \)[/tex]:
- [tex]\( f(3) = 5 \cdot f(2) = 5 \cdot 20 = 100 \)[/tex], matches [tex]\( f(3) = 100 \)[/tex].
### Option D:
#### [tex]\( f(1) = 5 \)[/tex]
#### [tex]\( f(n) = 4 \cdot f(n-1) \)[/tex] for [tex]\( n \geq 2 \)[/tex]
For [tex]\( n = 1 \)[/tex]:
- [tex]\( f(1) = 5 \)[/tex], does not match [tex]\( f(1) = 4 \)[/tex], so this option is incorrect.
### Conclusion
By checking each option against the terms of the given sequence, we find that only Option C produces the correct terms consistently.
Therefore, the correct recursive function that defines the sequence is:
[tex]\[ f(1) = 4 \][/tex]
[tex]\[ f(n) = 5 \cdot f(n-1) \text{ for } n \geq 2 \][/tex]
So, the correct answer is:
c. [tex]\( f(1) = 4, f(n) = 5 \cdot f(n-1) \text{ for } n \geq 2 \)[/tex]
