f(n) = 41 - 5n

Question

Complete the recursive formula of f(n).
f (1) =
f(n) = f(n - 1)+

Answer 1

To derive the recursive formula from the given function \( f(n) = 41 - 5n \), we need to express \( f(n) \) in terms of \( f(n-1) \).

1. Calculate \( f(n - 1) \): \[ f(n - 1) = 41 - 5(n - 1) = 41 - 5n + 5 = 46 - 5n \]

2. Find \( f(n) - f(n - 1) \): \[ f(n) = 41 - 5n \] Substituting \( f(n - 1) \): \[ f(n) - f(n - 1) = (41 - 5n) - (46 - 5n) = 41 - 5n - 46 + 5n = -5 \]

3. Construct the recursive formula: From the previous step, we can express \( f(n) \) in terms of \( f(n - 1) \): \[ f(n) = f(n - 1) - 5 \]

Thus, the recursive formula for \( f(n) \) is:

• \( f(1) = 41 - 5(1) = 36 \)
• \( f(n) = f(n - 1) - 5 \)

So the complete recursive formula is:

• \( f(1) = 36 \)
• \( f(n) = f(n - 1) - 5 \) for \( n > 1 \)