Let `alpha` and `beta` be the roots of `x^(2)-5x+3=0` with `alpha gt beta`. If `a_(n) = a^(n)-beta^(n)` for `n ge1`, then the value of `(3a_(6)+a_(8))/(a_(7))` is :

Options

A. 2
B. 3
C. 4
D. 5

Answer

To solve the problem, we will follow these steps:

1. **Identify the roots of the quadratic equation**: The given equation is \(x^2 - 5x + 3 = 0\). We will find the roots \(\alpha\) and \(\beta\) using the quadratic formula.

\[ \alpha, \beta = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \(a = 1\), \(b = -5\), and \(c = 3\).

\[ \alpha, \beta = \frac{5 \pm \sqrt{(-5)^2 - 4 \cdot 1 \cdot 3}}{2 \cdot 1} = \frac{5 \pm \sqrt{25 - 12}}{2} = \frac{5 \pm \sqrt{13}}{2} \]

Since \(\alpha > \beta\), we have:
\[ \alpha = \frac{5 + \sqrt{13}}{2}, \quad \beta = \frac{5 - \sqrt{13}}{2} \]

**Hint**: Use the quadratic formula to find the roots of the equation.

2. **Define the sequence \(a_n\)**: The sequence is defined as \(a_n = \alpha^n - \beta^n\) for \(n \geq 1\).

3. **Calculate \(a_6\), \(a_7\), and \(a_8\)**:
- \(a_6 = \alpha^6 - \beta^6\)
- \(a_7 = \alpha^7 - \beta^7\)
- \(a_8 = \alpha^8 - \beta^8\)

4. **Use the recurrence relation**: From the properties of the roots, we can derive a recurrence relation:
\[ a_n = 5a_{n-1} - 3a_{n-2} \]
This means:
- \(a_1 = \alpha - \beta\)
- \(a_2 = \alpha^2 - \beta^2\)
- Using the recurrence, we can compute \(a_3\), \(a_4\), \(a_5\), \(a_6\), \(a_7\), and \(a_8\).

5. **Calculate \(3a_6 + a_8\)**:
\[ 3a_6 + a_8 = 3(\alpha^6 - \beta^6) + (\alpha^8 - \beta^8) = 3\alpha^6 - 3\beta^6 + \alpha^8 - \beta^8 \]

6. **Factor out common terms**:
\[ = (3\alpha^6 + \alpha^8) - (3\beta^6 + \beta^8) = \alpha^6(3 + \alpha^2) - \beta^6(3 + \beta^2) \]

7. **Substitute the values into the expression**:
\[ \frac{3a_6 + a_8}{a_7} = \frac{\alpha^6(3 + \alpha^2) - \beta^6(3 + \beta^2)}{\alpha^7 - \beta^7} \]

8. **Simplify the expression**:
Using the recurrence relation, we find that:
\[ a_7 = \alpha^7 - \beta^7 = 5a_6 - 3a_5 \]
Thus, we can simplify the fraction.

9. **Final result**:
After substituting and simplifying, we find that:
\[ \frac{3a_6 + a_8}{a_7} = 5 \]

**Final Answer**: The value of \(\frac{3a_6 + a_8}{a_7}\) is \(5\).

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