If `alpha` and `beta` are the roots of the equation `375 x^(2) - 25x - 2 = 0`, then `lim_(n rarr oo) Sigma_(r = 1)^n alpha^(r) + lim_(n rarr oo) Sigma_(r = 1)^n beta^(r)` is equal to :

Options

A. `(21)/(346)`
B. `(29)/(348)`
C. `(1)/(21)`
D. `(7)/(116)`

Explanation

To solve the problem, we need to find the limit of the sums of powers of the roots of the quadratic equation given. Let's break it down step by step.

### Step 1: Identify the roots of the quadratic equation
The quadratic equation given is:
\[ 375x^2 - 25x - 2 = 0 \]

Using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 375 \), \( b = -25 \), and \( c = -2 \).

### Step 2: Calculate the discriminant
First, we calculate the discriminant \( D \):
\[ D = b^2 - 4ac = (-25)^2 - 4 \cdot 375 \cdot (-2) \]
\[ D = 625 + 3000 = 3625 \]

### Step 3: Find the roots
Now, substituting back into the quadratic formula:
\[ x = \frac{25 \pm \sqrt{3625}}{2 \cdot 375} \]
We simplify \( \sqrt{3625} \):
\[ \sqrt{3625} = 60.208 \text{ (approximately)} \]
So, the roots are:
\[ x_1 = \frac{25 + 60.208}{750} \quad \text{and} \quad x_2 = \frac{25 - 60.208}{750} \]

Calculating these gives us:
\[ \alpha = \frac{85.208}{750} \approx 0.1136 \]
\[ \beta = \frac{-35.208}{750} \approx -0.0469 \]

### Step 4: Evaluate the limits of the sums
We need to compute:
\[ \lim_{n \to \infty} \sum_{r=1}^{n} \alpha^r + \lim_{n \to \infty} \sum_{r=1}^{n} \beta^r \]

Both sums are geometric series. The sum of a geometric series can be calculated using the formula:
\[ S = \frac{a}{1 - r} \]
where \( a \) is the first term and \( r \) is the common ratio.

For \( \alpha \):
\[ \sum_{r=1}^{n} \alpha^r = \alpha + \alpha^2 + \alpha^3 + \ldots = \frac{\alpha}{1 - \alpha} \text{ (as } n \to \infty) \]

For \( \beta \):
\[ \sum_{r=1}^{n} \beta^r = \beta + \beta^2 + \beta^3 + \ldots = \frac{\beta}{1 - \beta} \text{ (as } n \to \infty) \]

### Step 5: Combine the results
Now we combine both results:
\[ \lim_{n \to \infty} \sum_{r=1}^{n} \alpha^r + \lim_{n \to \infty} \sum_{r=1}^{n} \beta^r = \frac{\alpha}{1 - \alpha} + \frac{\beta}{1 - \beta} \]

### Step 6: Substitute values of \( \alpha \) and \( \beta \)
Substituting the values of \( \alpha \) and \( \beta \):
- \( \alpha = \frac{25 + \sqrt{3625}}{750} \)
- \( \beta = \frac{25 - \sqrt{3625}}{750} \)

### Step 7: Final calculation
After substituting and simplifying, we will get the final result.