If `alpha, beta` are roots of equation `x^(2)-4x-3=0` and `s_(n)=alpha^(n)+beta^(n), n in N` then the value of `(s_(7)-4s_(6))/s_(5)` is

Options

A. 4
B. 3
C. 5
D. 7

Answer

To solve the problem, we need to find the value of \((s_7 - 4s_6)/s_5\) where \(s_n = \alpha^n + \beta^n\) and \(\alpha, \beta\) are the roots of the equation \(x^2 - 4x - 3 = 0\).

### Step 1: Find the roots \(\alpha\) and \(\beta\)

We start by solving the quadratic equation:
\[ x^2 - 4x - 3 = 0 \]
Using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):
- Here, \(a = 1\), \(b = -4\), and \(c = -3\).

Calculating the discriminant:
\[ b^2 - 4ac = (-4)^2 - 4 \cdot 1 \cdot (-3) = 16 + 12 = 28 \]

Now substituting back into the quadratic formula:
\[ x = \frac{4 \pm \sqrt{28}}{2} = \frac{4 \pm 2\sqrt{7}}{2} = 2 \pm \sqrt{7} \]
Thus, the roots are:
\[ \alpha = 2 + \sqrt{7}, \quad \beta = 2 - \sqrt{7} \]

### Step 2: Establish the recurrence relation for \(s_n\)

Using the roots, we can derive a recurrence relation for \(s_n\):
\[ s_n = \alpha^n + \beta^n \]
From the properties of roots, we have:
\[ s_n = 4s_{n-1} + 3s_{n-2} \]
This is derived from the characteristic equation of the roots.

### Step 3: Calculate \(s_5\), \(s_6\), and \(s_7\)

We need to find \(s_5\), \(s_6\), and \(s_7\). We can start with initial conditions:
- \(s_0 = 2\) (since \(\alpha^0 + \beta^0 = 1 + 1\))
- \(s_1 = 4\) (since \(\alpha + \beta = 4\))

Using the recurrence relation:
\[ s_2 = 4s_1 + 3s_0 = 4 \cdot 4 + 3 \cdot 2 = 16 + 6 = 22 \]
\[ s_3 = 4s_2 + 3s_1 = 4 \cdot 22 + 3 \cdot 4 = 88 + 12 = 100 \]
\[ s_4 = 4s_3 + 3s_2 = 4 \cdot 100 + 3 \cdot 22 = 400 + 66 = 466 \]
\[ s_5 = 4s_4 + 3s_3 = 4 \cdot 466 + 3 \cdot 100 = 1864 + 300 = 2164 \]
\[ s_6 = 4s_5 + 3s_4 = 4 \cdot 2164 + 3 \cdot 466 = 8656 + 1398 = 10054 \]
\[ s_7 = 4s_6 + 3s_5 = 4 \cdot 10054 + 3 \cdot 2164 = 40216 + 6492 = 46708 \]

### Step 4: Calculate \((s_7 - 4s_6)/s_5\)

Now we substitute \(s_5\), \(s_6\), and \(s_7\) into the expression:
\[ s_7 - 4s_6 = 46708 - 4 \cdot 10054 = 46708 - 40216 = 6492 \]
Now we calculate:
\[ \frac{s_7 - 4s_6}{s_5} = \frac{6492}{2164} \]
Calculating this gives:
\[ \frac{6492}{2164} = 3 \]

### Final Answer
Thus, the value of \((s_7 - 4s_6)/s_5\) is:
\[ \boxed{3} \]