If `alpha_(1), alpha_(2), alpha_(3), beta_(1), beta_(2), beta_(3)` are the values of n for which `sum_(r=0)^(n-1)x^(2r)` is divisible by `sum_(r=0)^(n-1)x^(r )`, then the triangle having vertices `(alpha_(1), beta_(1)),(alpha_(2),beta_(2))` and `(alpha_(3), beta_(3))` cannot be (Option 1) an isosceles triangle Option 2) a right angled isosceles triangle Option 3) a right angled triangle Option 4) an equilateral triangle

Options

A. an isosceles triangle
B. a right angled isosceles triangle
C. a right angled triangle
D. an equilateral triangle

Answer

To solve the problem, we need to analyze the conditions under which the summation \( \sum_{r=0}^{n-1} x^{2r} \) is divisible by \( \sum_{r=0}^{n-1} x^{r} \).

### Step 1: Write the Summations
The summation \( \sum_{r=0}^{n-1} x^{2r} \) can be expressed as:
\[ \sum_{r=0}^{n-1} x^{2r} = 1 + x^2 + x^4 + \ldots + x^{2(n-1)} = \frac{1 - x^{2n}}{1 - x^2} \]
Similarly, the summation \( \sum_{r=0}^{n-1} x^{r} \) can be expressed as:
\[ \sum_{r=0}^{n-1} x^{r} = 1 + x + x^2 + \ldots + x^{n-1} = \frac{1 - x^n}{1 - x} \]

### Step 2: Set Up the Divisibility Condition
We want to find values of \( n \) such that:
\[ \frac{1 - x^{2n}}{1 - x^2} \text{ is divisible by } \frac{1 - x^n}{1 - x} \]
This means that:
\[ (1 - x^{2n})(1 - x) \text{ must be divisible by } (1 - x^2)(1 - x^n) \]

### Step 3: Analyze the Condition
For the above divisibility to hold, we can analyze the roots of the equations. The roots of \( 1 - x^{2n} = 0 \) are the \( 2n \)-th roots of unity, and the roots of \( 1 - x^n = 0 \) are the \( n \)-th roots of unity.

### Step 4: Determine Values of \( n \)
From the analysis, it can be concluded that \( n \) must be odd. The values of \( n \) can be \( 1, 3, 5, 7, 9, 11, \ldots \).

### Step 5: Assign Values to \( \alpha \) and \( \beta \)
Let us assign:
- \( \alpha_1 = 1 \), \( \alpha_2 = 3 \), \( \alpha_3 = 5 \)
- \( \beta_1 = 7 \), \( \beta_2 = 9 \), \( \beta_3 = 11 \)

The vertices of the triangle are:
- \( (1, 7) \)
- \( (3, 9) \)
- \( (5, 11) \)

### Step 6: Calculate the Sides of the Triangle
Using the distance formula, we calculate the lengths of the sides of the triangle formed by these vertices:
1. Distance \( AB \) between points \( A(1, 7) \) and \( B(3, 9) \):
\[ AB = \sqrt{(3 - 1)^2 + (9 - 7)^2} = \sqrt{2^2 + 2^2} = \sqrt{8} = 2\sqrt{2} \]
2. Distance \( BC \) between points \( B(3, 9) \) and \( C(5, 11) \):
\[ BC = \sqrt{(5 - 3)^2 + (11 - 9)^2} = \sqrt{2^2 + 2^2} = \sqrt{8} = 2\sqrt{2} \]
3. Distance \( CA \) between points \( C(5, 11) \) and \( A(1, 7) \):
\[ CA = \sqrt{(5 - 1)^2 + (11 - 7)^2} = \sqrt{4^2 + 4^2} = \sqrt{32} = 4\sqrt{2} \]

### Step 7: Classify the Triangle
From the calculated side lengths:
- \( AB = 2\sqrt{2} \)
- \( BC = 2\sqrt{2} \)
- \( CA = 4\sqrt{2} \)

Since two sides are equal and one side is different, this triangle is an isosceles triangle. However, it cannot be equilateral since all sides are not equal.

### Conclusion
Thus, the triangle with vertices \( (1, 7), (3, 9), (5, 11) \) cannot be an equilateral triangle.

**Final Answer: Option 4 - an equilateral triangle.**