### Sequences

Rich example

A ball is thrown horizontally at a speed of $\quantity{2}{m\,s^{-1}}$ from a point $\quantity{1.25}{m}$ above $O$ which is on a flat horizontal surface. After each bounce it reaches a height which is $75\%$ of its maximum height after the previous bounce. Ignore air resistance and take the acceleration of gravity, $g$, to be $\quantity{10}{m\,s^{-2}}$.
• Calculate $t_n$, the time the ball is in the air between the $n^\text{th}$ and $(n+1)^\text{th}$ bounces.
• How far from $O$ does the ball get before it stops bouncing?