To find an explicit formula for \( f(n) \), we start with the given recursion:
- Initial condition: \( f(1) = -62 \)
- Recurrence relation: \( f(n) = f(n - 1) + 5 \)
We can derive the explicit formula by calculating the first few terms based on this recurrence relation:
- \( f(1) = -62 \)
- \( f(2) = f(1) + 5 = -62 + 5 = -57 \)
- \( f(3) = f(2) + 5 = -57 + 5 = -52 \)
- \( f(4) = f(3) + 5 = -52 + 5 = -47 \)
It appears that each time we move to the next value of \( n \), we increase the value of \( f(n) \) by 5.
The sequence can be expressed more generally. Starting from \( f(1) \) and adding 5 for each increment in \( n \),
\[ f(n) = f(1) + 5(n - 1) \]
Substituting in the initial condition:
\[ f(n) = -62 + 5(n - 1) \]
Now simplifying,
\[ f(n) = -62 + 5n - 5 \] \[ f(n) = 5n - 67 \]
Thus, the explicit formula for \( f(n) \) is:
\[ \boxed{f(n) = 5n - 67} \]