Let `alpha and beta` are the roots of `x^2 – x – 1 = 0` such that `P_k = alpha^k + beta^k , k ge 1` then which one is incorrect?

Options

A. `P_5 = P_3 xx P_2`
B. `P_1+P_2+P_3+P_4+P_5=26`
C. `P_3=P_5-P_4`
D. `P_5=11`

Explanation

To solve the problem, we need to find the values of \( P_k = \alpha^k + \beta^k \) for the roots \( \alpha \) and \( \beta \) of the equation \( x^2 - x - 1 = 0 \).

### Step 1: Find the roots \( \alpha \) and \( \beta \)
The roots of the quadratic equation \( x^2 - x - 1 = 0 \) can be found using the quadratic formula:
\[ \alpha, \beta = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1, b = -1, c = -1 \):
\[ \alpha, \beta = \frac{1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot (-1)}}{2 \cdot 1} = \frac{1 \pm \sqrt{5}}{2} \]
Thus, we have:
\[ \alpha = \frac{1 + \sqrt{5}}{2}, \quad \beta = \frac{1 - \sqrt{5}}{2} \]

### Step 2: Calculate \( P_1 \)
Using the definition of \( P_k \):
\[ P_1 = \alpha^1 + \beta^1 = \alpha + \beta \]
From Vieta's formulas, we know:
\[ \alpha + \beta = 1 \]
Thus,
\[ P_1 = 1 \]

### Step 3: Calculate \( P_2 \)
Using the relation \( P_k = \alpha^k + \beta^k \):
\[ P_2 = \alpha^2 + \beta^2 \]
Using the identity \( \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \):
\[ \alpha\beta = -1 \quad \text{(from Vieta's)} \]
So,
\[ P_2 = 1^2 - 2(-1) = 1 + 2 = 3 \]

### Step 4: Calculate \( P_3 \)
Using the recurrence relation \( P_k = P_{k-1} + P_{k-2} \):
\[ P_3 = P_2 + P_1 = 3 + 1 = 4 \]

### Step 5: Calculate \( P_4 \)
\[ P_4 = P_3 + P_2 = 4 + 3 = 7 \]

### Step 6: Calculate \( P_5 \)
\[ P_5 = P_4 + P_3 = 7 + 4 = 11 \]

### Step 7: Check the options for correctness
We have calculated:
- \( P_1 = 1 \)
- \( P_2 = 3 \)
- \( P_3 = 4 \)
- \( P_4 = 7 \)
- \( P_5 = 11 \)

Now, we check the statements given in the options. We need to find which one is incorrect.

1. **Check if \( P_5 = 11 \)**: Correct.
2. **Check if \( P_3 = 4 \)**: Correct.
3. **Check if \( P_4 + P_5 = 26 \)**: \( 7 + 11 = 18 \) (Incorrect).
4. **Check if \( 3P_3 = 12 \)**: \( 3 \times 4 = 12 \) (Correct).

### Conclusion
The incorrect statement is the one that claims \( P_4 + P_5 = 26 \).