In a arithmetic progression whose first term is `alpha` and common difference is ` beta , alpha,beta ne 0 ` the ratio r of the sum of the first n terms to the sum of n terms succeending them, does not depend on n. Then which of the following is /are correct ?

Options

A. `alpha: beta = 2:1`
B. If `alpha " and " beta ` are roots of the equation `ax^2+bx+c =0` then `2 b^2 = 9ac`
C. The sum of infinite `G.P 1+r+r^2 + …. Is 3//2`
D. If `alpha =1` , then sum of 10 terms of A.P is 100

Explanation

To solve the problem, we need to analyze the given arithmetic progression (AP) and the relationship between the sums of its terms.

### Step-by-Step Solution:

1. **Define the terms of the AP**:
Let the first term of the AP be \( \alpha \) and the common difference be \( \beta \). The \( n \)-th term of the AP can be expressed as:
\[ a_n = \alpha + (n-1)\beta \]

2. **Sum of the first n terms**:
The sum of the first \( n \) terms \( S_n \) of an AP is given by the formula:
\[ S_n = \frac{n}{2} \left(2\alpha + (n-1)\beta\right) \]

3. **Sum of the first 2n terms**:
The sum of the first \( 2n \) terms \( S_{2n} \) can be calculated similarly:
\[ S_{2n} = \frac{2n}{2} \left(2\alpha + (2n-1)\beta\right) = n \left(2\alpha + (2n-1)\beta\right) \]

4. **Calculate the ratio \( r \)**:
The ratio \( r \) of the sum of the first \( n \) terms to the sum of the next \( n \) terms (which is \( S_{2n} - S_n \)) can be expressed as:
\[ r = \frac{S_n}{S_{2n} - S_n} \]
Here, \( S_{2n} - S_n = S_{2n} - S_n = n \left(2\alpha + (2n-1)\beta\right) - \frac{n}{2} \left(2\alpha + (n-1)\beta\right) \).

5. **Simplifying the expression**:
Substitute the expressions for \( S_n \) and \( S_{2n} \):
\[ S_{2n} - S_n = n \left(2\alpha + (2n-1)\beta\right) - \frac{n}{2} \left(2\alpha + (n-1)\beta\right) \]
Simplifying this will yield:
\[ S_{2n} - S_n = n \left(2\alpha + 2n\beta - \beta\right) - \frac{n}{2} \left(2\alpha + n\beta - \beta\right) \]

6. **Setting the ratio independent of \( n \)**:
For the ratio \( r \) to be independent of \( n \), the coefficients of \( n \) in the numerator and denominator must cancel out. This leads to the condition:
\[ 2\alpha - \beta = 0 \quad \Rightarrow \quad 2\alpha = \beta \]

7. **Finding the relationship between \( \alpha \) and \( \beta \)**:
From \( 2\alpha = \beta \), we can express \( \alpha \) in terms of \( \beta \):
\[ \alpha = \frac{\beta}{2} \]

8. **Conclusion**:
The ratio \( r \) can be calculated using the derived relationship. The conditions given in the problem lead to the conclusion that \( \alpha \) and \( \beta \) have a specific ratio, and we can check the options provided in the question to find which are correct based on this relationship.