A function f is defined such that `f(1)=2, f(2)=5, and f(n)=f(n-1)-f(n-2)` for all integer values of n greater than 2. What is the value of f(4)?

Options

A. `-8`
B. `-2`
C. `2`
D. ``

Answer

To find the value of \( f(4) \) given the function defined by the rules:

1. \( f(1) = 2 \)
2. \( f(2) = 5 \)
3. \( f(n) = f(n-1) - f(n-2) \) for all integers \( n > 2 \)

we will follow these steps:

### Step 1: Find \( f(3) \)
Using the recursive formula for \( n = 3 \):
\[ f(3) = f(3-1) - f(3-2) = f(2) - f(1) \]
Substituting the known values:
\[ f(3) = f(2) - f(1) = 5 - 2 = 3 \]

### Step 2: Find \( f(4) \)
Now, we can find \( f(4) \) using the recursive formula for \( n = 4 \):
\[ f(4) = f(4-1) - f(4-2) = f(3) - f(2) \]
Substituting the values we have:
\[ f(4) = f(3) - f(2) = 3 - 5 = -2 \]

### Final Answer
Thus, the value of \( f(4) \) is:
\[ \boxed{-2} \]