After each collision speed becomes $e$ times

$\therefore$ Speed after $1^{st}$ collision is ${\nu }_1=e{\nu }_0$

$\therefore$ Maximum height after $1^{st}$ collision is,

$h_1=\frac{{\nu }_1^2}{2g}=e^2\left(\frac{{\nu }_0^2}{2g}\right)=e^{2}h$

Similarly maximum height after $2^{nd}$ collision is,

$h_2=e^2{h}_1=e^{4}h$

maximum height after $3^{rd}$ collision is,

$h_3=e^2{h}_2=e^{6}h$

$\therefore$ The height to which the ball goes up after it rebounds for the $n^{th}$ time is $h_n=e^{2n}h$