Question

\begin{cases}
f(1)=-8
\\\\
f(n)=f(n-1)-3
\end{cases}

Answer 1

To solve the recurrence relation defined by the cases you provided, we can break it down step by step.

The initial condition is:
- \( f(1) = -8 \)

The recurrence relation is:
- \( f(n) = f(n-1) - 3 \)

This means that each term in the sequence is obtained by subtracting 3 from the previous term.

Let's compute the first few values:
- \( f(2) = f(1) - 3 = -8 - 3 = -11 \)
- \( f(3) = f(2) - 3 = -11 - 3 = -14 \)
- \( f(4) = f(3) - 3 = -14 - 3 = -17 \)
- \( f(5) = f(4) - 3 = -17 - 3 = -20 \)

From this pattern, we can observe that as \( n \) increases, \( f(n) \) decreases by 3 for each increment in \( n \).

We can generalize this to express \( f(n) \) in terms of \( n \):
Starting from the base case:
\[
f(n) = f(1) - 3(n - 1)
\]
Substituting \( f(1) = -8 \):
\[
f(n) = -8 - 3(n - 1)
\]
Now simplify this:
\[
f(n) = -8 - 3n + 3
\]
\[
f(n) = -3n - 5
\]

So the closed-form expression for \( f(n) \) is:
\[
f(n) = -3n - 5
\]

This formula allows us to compute \( f(n) \) for any positive integer \( n \).