Question

\begin{cases}
f(1)=-71
\\\\
f(n)=f(n-1)\cdot 4.2
\end{cases}

Answer 1

The function \( f(n) \) is defined recursively, starting with the initial condition \( f(1) = -71 \) and the recursive formula \( f(n) = f(n-1) \cdot 4.2 \).

Let's compute a few values to better understand this function:

1. **Calculate \( f(2) \)**:
\[
f(2) = f(1) \cdot 4.2 = -71 \cdot 4.2 = -298.2
\]

2. **Calculate \( f(3) \)**:
\[
f(3) = f(2) \cdot 4.2 = -298.2 \cdot 4.2 = -1252.428
\]

3. **Calculate \( f(4) \)**:
\[
f(4) = f(3) \cdot 4.2 = -1252.428 \cdot 4.2 = -5264.1956
\]

4. **Calculate \( f(5) \)**:
\[
f(5) = f(4) \cdot 4.2 = -5264.1956 \cdot 4.2 = -22105.61752
\]

Based on this pattern, we can derive a general formula for \( f(n) \).

### General Formula
Each term builds on the previous one by multiplying by \( 4.2 \). Thus, we can express \( f(n) \) in terms of \( f(1) \):

\[
f(n) = f(1) \cdot (4.2)^{n-1}
\]

Substituting \( f(1) = -71 \):

\[
f(n) = -71 \cdot (4.2)^{n-1}
\]

### Conclusion
The function \( f(n) \) for any integer \( n \geq 1 \) is given by:

\[
f(n) = -71 \cdot (4.2)^{n-1}
\]