Let ` alpha and beta` be the roots of the equation ` x^(2) -px+q =0 and V_(n) = alpha^(n) + beta^(n)` , Show that ` V_(n+1) = pV_(n) -qV_(n-1)` find ` V_(5)`

Answer

To solve the problem, we need to show that \( V_{n+1} = pV_n - qV_{n-1} \) and then find \( V_5 \).

### Step 1: Understand the Roots
Given the quadratic equation:
\[ x^2 - px + q = 0 \]
Let \( \alpha \) and \( \beta \) be the roots of this equation. By Vieta's formulas, we know:
\[ \alpha + \beta = p \quad \text{(sum of roots)} \]
\[ \alpha \beta = q \quad \text{(product of roots)} \]

### Step 2: Define \( V_n \)
We define:
\[ V_n = \alpha^n + \beta^n \]

### Step 3: Prove the Recurrence Relation
We need to show that:
\[ V_{n+1} = pV_n - qV_{n-1} \]

**Left-Hand Side (LHS)**:
\[ V_{n+1} = \alpha^{n+1} + \beta^{n+1} \]

**Right-Hand Side (RHS)**:
Using the definitions of \( V_n \) and \( V_{n-1} \):
\[ pV_n = p(\alpha^n + \beta^n) = (\alpha + \beta)(\alpha^n + \beta^n) = \alpha^{n+1} + \beta^{n+1} + \alpha^n \beta + \beta^n \alpha \]
\[ qV_{n-1} = q(\alpha^{n-1} + \beta^{n-1}) = \alpha \beta (\alpha^{n-1} + \beta^{n-1}) = \alpha^{n} \beta + \beta^{n} \alpha \]
Thus,
\[ pV_n - qV_{n-1} = (\alpha^{n+1} + \beta^{n+1} + \alpha^n \beta + \beta^n \alpha) - (\alpha^{n} \beta + \beta^{n} \alpha) \]
The terms \( \alpha^n \beta \) and \( \beta^n \alpha \) cancel out, leading to:
\[ pV_n - qV_{n-1} = \alpha^{n+1} + \beta^{n+1} = V_{n+1} \]
Thus, we have shown that:
\[ V_{n+1} = pV_n - qV_{n-1} \]

### Step 4: Calculate \( V_5 \)
To find \( V_5 \), we need \( V_4 \) and \( V_3 \).

1. **Calculate \( V_3 \)**:
\[ V_3 = pV_2 - qV_1 \]
where:
\[ V_1 = \alpha + \beta = p \]
\[ V_2 = \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta = p^2 - 2q \]
Thus,
\[ V_3 = p(p^2 - 2q) - q(p) = p^3 - 2pq - qp = p^3 - 3pq \]

2. **Calculate \( V_4 \)**:
\[ V_4 = pV_3 - qV_2 \]
Substituting \( V_3 \) and \( V_2 \):
\[ V_4 = p(p^3 - 3pq) - q(p^2 - 2q) \]
\[ = p^4 - 3p^2q - qp^2 + 2q^2 = p^4 - 4p^2q + 2q^2 \]

3. **Finally, calculate \( V_5 \)**:
\[ V_5 = pV_4 - qV_3 \]
Substituting \( V_4 \) and \( V_3 \):
\[ V_5 = p(p^4 - 4p^2q + 2q^2) - q(p^3 - 3pq) \]
\[ = p^5 - 4p^3q + 2pq^2 - qp^3 + 3q^2p \]
\[ = p^5 - 5p^3q + 5pq^2 \]

### Final Result
Thus, we have:
\[ V_5 = p^5 - 5p^3q + 5pq^2 \]