Suppose `alpha, beta` are roots of `8x^(2) - 10 x + 3 = 0` , then `sum_(n=0) ^(infty) (alpha^(n) + beta^(n))` is

Options

A. `7//4`
B. `3//7`
C. 6
D. 7

Answer

To solve the problem, we need to find the sum of the series \( S = \sum_{n=0}^{\infty} (\alpha^n + \beta^n) \), where \( \alpha \) and \( \beta \) are the roots of the quadratic equation \( 8x^2 - 10x + 3 = 0 \).

### Step-by-Step Solution:

1. **Identify the roots of the quadratic equation**:
The roots \( \alpha \) and \( \beta \) can be found using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 8 \), \( b = -10 \), and \( c = 3 \).

\[ \alpha, \beta = \frac{10 \pm \sqrt{(-10)^2 - 4 \cdot 8 \cdot 3}}{2 \cdot 8} \]
\[ = \frac{10 \pm \sqrt{100 - 96}}{16} = \frac{10 \pm \sqrt{4}}{16} = \frac{10 \pm 2}{16} \]
\[ = \frac{12}{16}, \frac{8}{16} = \frac{3}{4}, \frac{1}{2} \]
Thus, \( \alpha = \frac{3}{4} \) and \( \beta = \frac{1}{2} \).

2. **Express the series**:
The series can be split into two parts:
\[ S = \sum_{n=0}^{\infty} \alpha^n + \sum_{n=0}^{\infty} \beta^n \]

3. **Sum of the geometric series**:
The sum of an infinite geometric series \( \sum_{n=0}^{\infty} r^n \) is given by:
\[ S = \frac{1}{1 - r} \quad \text{(for } |r| < 1\text{)} \]
Here, both \( \alpha \) and \( \beta \) are less than 1, so we can apply this formula.

For \( \alpha = \frac{3}{4} \):
\[ S_\alpha = \sum_{n=0}^{\infty} \left(\frac{3}{4}\right)^n = \frac{1}{1 - \frac{3}{4}} = \frac{1}{\frac{1}{4}} = 4 \]

For \( \beta = \frac{1}{2} \):
\[ S_\beta = \sum_{n=0}^{\infty} \left(\frac{1}{2}\right)^n = \frac{1}{1 - \frac{1}{2}} = \frac{1}{\frac{1}{2}} = 2 \]

4. **Combine the results**:
Now, we add the two sums:
\[ S = S_\alpha + S_\beta = 4 + 2 = 6 \]

### Final Answer:
Thus, the sum \( \sum_{n=0}^{\infty} (\alpha^n + \beta^n) = 6 \).

---