If f (0) = 2, f (1) = 3 and f (x+2) =2f (x) - f (x + 1) for every real x then f (5) is equal to

Options

A. 13
B. 1
C. 5
D. 7

Answer

To find \( f(5) \) given the conditions \( f(0) = 2 \), \( f(1) = 3 \), and the recurrence relation \( f(x+2) = 2f(x) - f(x+1) \), we will calculate the values of \( f(2) \), \( f(3) \), \( f(4) \), and finally \( f(5) \).

### Step 1: Calculate \( f(2) \)
Using the recurrence relation with \( x = 0 \):
\[ f(2) = 2f(0) - f(1) \]
Substituting the known values:
\[ f(2) = 2 \cdot 2 - 3 = 4 - 3 = 1 \]

### Step 2: Calculate \( f(3) \)
Now, using the recurrence relation with \( x = 1 \):
\[ f(3) = 2f(1) - f(2) \]
Substituting the known values:
\[ f(3) = 2 \cdot 3 - 1 = 6 - 1 = 5 \]

### Step 3: Calculate \( f(4) \)
Next, using the recurrence relation with \( x = 2 \):
\[ f(4) = 2f(2) - f(3) \]
Substituting the known values:
\[ f(4) = 2 \cdot 1 - 5 = 2 - 5 = -3 \]

### Step 4: Calculate \( f(5) \)
Finally, using the recurrence relation with \( x = 3 \):
\[ f(5) = 2f(3) - f(4) \]
Substituting the known values:
\[ f(5) = 2 \cdot 5 - (-3) = 10 + 3 = 13 \]

Thus, the value of \( f(5) \) is \( \boxed{13} \).