In an arithmetic progression whose first term is $\mathrm{\alpha }$ and common difference is $\mathrm{\beta }; \mathrm{\alpha }, \mathrm{\beta }\ne 0$, the ratio r of the sum of the first n terms to the sum of n terms succeeding them, does not depend on n. Then which of the following is/are correct?

Let S_n denote the sum of first n terms of A.P..

According to the question,

$\mathrm{r}=\frac{{\mathrm{S}}_{\mathrm{n}}}{{\mathrm{S}}_{2\mathrm{n}}-{\mathrm{S}}_{\mathrm{n}}}=\frac{1}{\frac{{\mathrm{S}}_{2\mathrm{n}}}{{\mathrm{S}}_{\mathrm{n}}}-1}$ is independent of n.

Therefore, $\frac{{\mathrm{S}}_{2\mathrm{n}}}{{\mathrm{S}}_{\mathrm{n}}}$ is independent of n.

Now, $\frac{{\mathrm{S}}_{2\mathrm{n}}}{{\mathrm{S}}_{\mathrm{n}}}=\frac{\frac{2\mathrm{n}}{2}(2\mathrm{\alpha }+(2\mathrm{n}-1\left)\mathrm{\beta }\right)}{\frac{\mathrm{n}}{2}(2\mathrm{\alpha }+(\mathrm{n}-1\left)\mathrm{\beta }\right)}=\frac{2(2\mathrm{\alpha }-\mathrm{\beta }+2\mathrm{n\beta })}{(2\mathrm{\alpha }-\mathrm{\beta }+\mathrm{n\beta })}$

This ratio is independent of n if

$2\mathrm{\alpha }-\mathrm{\beta }=0$

$\begin{array}{l}\therefore \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{\alpha }:\mathrm{\beta }=1:2\\ \therefore \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{r}=\frac{1}{4-1}=\frac{1}{3}\end{array}$

$\mathrm{\alpha }\text{ and }\mathrm{\beta }$ are the roots of the equation ${\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c}=0$.

$\therefore 2\mathrm{\alpha }+\mathrm{\alpha }=-\frac{\mathrm{b}}{\mathrm{a}}\text{ and }2\mathrm{\alpha }\times \mathrm{\alpha }=\frac{\mathrm{c}}{\mathrm{a}}$

Solving these, we get

$2{\left(-\frac{\mathrm{b}}{3\mathrm{a}}\right)}^2=\frac{\mathrm{c}}{\mathrm{a}}$

$\Rightarrow 2{\mathrm{b}}^2=9\mathrm{ac}$

$1+\mathrm{r}+{\mathrm{r}}^2+\dots =\frac{1}{1-\mathrm{r}}=\frac{1}{1-\frac{1}{3}}=\frac{3}{2}$

$\mathrm{\alpha }=1\text{ then }\mathrm{\beta }=2$

$\therefore \text{ Sum of }10\text{ terms A.P. }=\frac{10}{2}(2\times 1+(10-1)\times 2)=100$