Q.A ball is dropped from a height of $100$ m and rebounds each time to $\dfrac34$ of the height from which it fell. Find the total distance it travels before coming to rest.

[!TLDR] Total distance $=h+2\cdot\dfrac{hr}{1-r}$ (infinite G.P. of up-and-down bounces) with $h=100,r= …

The ball first falls $h=100$ m. After that, every bounce goes up to $\tfrac34$ of the previous height and then falls back down through the same distance — so after the first fall, each rebound height contributes twice (once up, once down) to the total distance travelled, and the successive rebound heights $hr, hr^2, hr^3,\ldots$ form an infinite G.P. with first term $hr$ and ratio $r=\tfrac34$ (with $|r|<1$, so it sums to a finite value). Total distance $=h + 2\left(hr+hr^2+hr^3+\cdots\right) …

Separate the initial fall (counted once) from every subsequent rebound (counted twice, up and down); recognize the rebound heights as an infinite G.P. …

Forgetting to double the rebound-height sum (since each rebound contributes both an upward and a downward distance) is the class …