Let `alpha,beta` and `gamma` be the roots of the equation `x^(3)-x-1=0`. If `P_(k)=(alpha)^(k)+(beta)^(k)+(gamma)^(k),kge1`, then which one of the following statements is not true?

Options

A. `P_(3)=P_(1)+P_(2)+1`
B. `P_(2)=P_(4)`
C. `P_(5)=P_(6)`
D. `P_(5)=P_(3)`

Explanation

To solve the problem, we need to analyze the roots of the cubic equation \( x^3 - x - 1 = 0 \) and calculate the values of \( P_k = \alpha^k + \beta^k + \gamma^k \) for various values of \( k \). Here are the steps to find the solution:

### Step 1: Identify the roots and their properties
The roots of the equation \( x^3 - x - 1 = 0 \) are denoted as \( \alpha, \beta, \gamma \). Using Vieta's formulas, we can establish the following relationships:
- \( \alpha + \beta + \gamma = 0 \)
- \( \alpha\beta + \beta\gamma + \gamma\alpha = -1 \)
- \( \alpha\beta\gamma = 1 \)

### Step 2: Calculate \( P_1 \)
We can calculate \( P_1 \):
\[ P_1 = \alpha + \beta + \gamma = 0 \]

### Step 3: Calculate \( P_2 \)
To find \( P_2 \), we use the identity:
\[ P_2 = \alpha^2 + \beta^2 + \gamma^2 = (\alpha + \beta + \gamma)^2 - 2(\alpha\beta + \beta\gamma + \gamma\alpha) \]
Substituting the known values:
\[ P_2 = 0^2 - 2(-1) = 2 \]

### Step 4: Calculate \( P_3 \)
For \( P_3 \), we can use the fact that \( \alpha^3 = \alpha + 1 \) (since \( \alpha \) is a root of the equation):
\[ P_3 = \alpha^3 + \beta^3 + \gamma^3 = (\alpha + 1) + (\beta + 1) + (\gamma + 1) = (\alpha + \beta + \gamma) + 3 = 0 + 3 = 3 \]

### Step 5: Establish a recurrence relation
We can derive a recurrence relation for \( P_k \):
\[ P_k = P_{k-2} + P_{k-1} \]
This is derived from the polynomial relation of the roots.

### Step 6: Calculate \( P_4 \) and \( P_5 \)
Using the recurrence relation:
- For \( P_4 \):
\[ P_4 = P_2 + P_1 = 2 + 0 = 2 \]
- For \( P_5 \):
\[ P_5 = P_3 + P_2 = 3 + 2 = 5 \]

### Step 7: Calculate \( P_6 \)
Continuing with the recurrence:
\[ P_6 = P_4 + P_3 = 2 + 3 = 5 \]

### Step 8: Summary of values
Now we have:
- \( P_1 = 0 \)
- \( P_2 = 2 \)
- \( P_3 = 3 \)
- \( P_4 = 2 \)
- \( P_5 = 5 \)
- \( P_6 = 5 \)

### Step 9: Evaluate the options
Now we check the statements:
1. \( P_3 = P_2 + P_1 + 1 \) → \( 3 = 2 + 0 + 1 \) (True)
2. \( P_2 = P_4 \) → \( 2 = 2 \) (True)
3. \( P_5 = P_6 \) → \( 5 = 5 \) (True)
4. \( P_5 = P_3 \) → \( 5 \neq 3 \) (False)

### Conclusion
The statement that is **not true** is:
**Option 4: \( P_5 = P_3 \)**.