Question

Suppose that you drop a ball from a window 50 metres above the ground. The ball bounces to 50% of its previous height with each bounce. If the ball continues to bounce in this manner how far will it have traveled, up and down, from the time it was dropped from the window until the 25th bounce?
200 m
100 m
75 m
150 m

Answer 1

We have been given that  you drop a ball from a window 50 metres above the ground. The ball bounces to 50% of its previous height with each bounce. We are asked to find the total distance traveled by up and down from the time it was dropped from the window until the 25th bounce.            

We will use sum of geometric sequence formula to solve our given problem.

$S_n=\frac{a\cdot(1-r^n)}{1-r}$, where,

a = First term of sequence,

r = Common ratio,

n = Number of terms.

For our given problem $a=50$, $r=50\%=\frac{50}{100}=0.5$ and $n=25$.

$S_{25}=\frac{50\cdot(1-(0.5)^{25})}{1-0.5}$

$S_{25}=\frac{50\cdot(1-0.0000000298023224)}{0.5}$

$S_{25}=\frac{50\cdot(0.9999999701976776)}{0.5}$

$S_{25}=100\cdot(0.9999999701976776)$

$S_{25}=99.99999701976776\approx 100$

Therefore, the ball will travel 100 meters and option B is the correct choice.