Question

f(n)=64+6nComplete the recursive formula of
\[f(n)\].
\[f(1)=\]

Answer 1

To formulate the recursive definition for \( f(n) = 64 + 6n \), we can start by determining the base case and the recursive relationship.

1. **Base case**: We need to calculate \( f(1) \).
\[
f(1) = 64 + 6(1) = 64 + 6 = 70
\]

2. **Recursive relationship**: To find a recursion, we need to find a relationship between \( f(n) \) and \( f(n-1) \).
\[
f(n) = 64 + 6n
\]
\[
f(n-1) = 64 + 6(n-1) = 64 + 6n - 6 = 64 + 6n - 6 = f(n) - 6
\]
Therefore, we can express \( f(n) \) in terms of \( f(n-1) \):
\[
f(n) = f(n-1) + 6
\]

Putting this all together, the recursive formula for \( f(n) \) is:

\[
\begin{aligned}
f(1) & = 70 \\
f(n) & = f(n-1) + 6 \quad \text{for } n > 1
\end{aligned}
\]

Thus, the complete recursive formula is specified with \( f(1) = 70 \) as the base case.