If tangent at a point $P_1$ (other than (0,0)) on the curve $y^2=a{x}^3$ meets the curve again at  $P_2$. The tangent at$P_2$  meets the curve again at  $P_3$ and so on, then $\underset{n\to \infty }{\lim }\sum _{i=1}^n{x}_i,$ where $x_i\text{'}s$ are abscissa of $P_i$  with  $x_1=3$ is

Let   $P_1=\left(t_1^2,\sqrt{a}{t}_1^3\right)$   and  $P_2=\left(t_2^2,\sqrt{a}{t}_2^3\right)$

$\because 2y\frac{dy}{dx}=3a{x}^2\Rightarrow \frac{dy}{dx}=\frac{3a{x}^2}{2y}$

Slope of tangent at $P_1=\frac{3\sqrt{a}{t}_1}{2}=\frac{\sqrt{a}\left(t_1^3-t_2^3\right)}{\left(t_1^2-t_2^2\right)}$

$\Rightarrow 3\left(t_1\right)\left(t_1+t_2\right)=2\left(t_1^2+t_2^2+2t_1{t}_2\right)$

$\Rightarrow t_1^2-t_2^2+t_1{t}_2=0\Rightarrow \left(t_1+2t_2\right)\left(t_1-t_2\right)=0$

$t_1=-2t_2$

$\therefore$ Abscissae, $t_1^2,t_2^2,t_3^2,\mathrm{...}\Rightarrow t_1^2,\frac{t_1^2}{4},\frac{t_1^2}{16},\mathrm{...}$ will be G.P. with common ratio $\frac{1}{4}$ for $\forall a$

$\therefore$ sum of abscissa = $\frac{x_1}{1-\frac{1}{4}}=\frac{3}{3/4}=4$