Lets consider quadratic equation $ax^2 + bx + c = 0$ where $a, b, c \in R$ and $a\ne 0$. If above equation has roots $\alpha, \beta$, then $\displaystyle{\alpha + \beta = -\frac{b}{a} , \alpha\beta = \frac{c}{a}}$ and the equation can be written as $ax^2+ bx + c = a(x - \alpha)(x - \beta)$. Also, if $a_1, a_2, a_3, a_4$, ..... are in A.P., then $a_2 - a_1 = a_3 - a_2 = a_4 - a_3 = ... \ne 0$ and if $b_1, b_2, b_3, b_4$, ... are in G.P., then $\displaystyle{\frac{b_2}{b_1} = \frac{b_3}{b_2} = \frac{b_4}{b_3}} =$ ... $\ne1$ Now if $c_1$, $c_2$, $c_3$, $c_4$, ... are in HP, then $\displaystyle{\frac{1}{c_2} - \frac{1}{c_1} = \frac{1}{c_3} - \frac{1}{c_2} = \frac{1}{c_4} - \frac{1}{c_3}} = $... $\ne 0$. If the roots of equation $a(b - c)x^2 + b(c - a)x + c(a - b) = 0$ are equal, then $a, b, c$ are in