Geometric Sequences

Question

A ball is dropped from a height of 4 meters. On each upward bounce the ball returns to 1/2 of its previous height. Find the total vertical distance the ball travels before coming to rest. Round to the nearest meter as needed.

Answer 1

Limit of the sum of a sequence: 8+4+2+1+1/2+1/4+1/8......= 16 meters exactly

8+4+2+1=15 the remaining fractions are similar to Zeno's Paradox, about how a man can never reach his destination because he keeps going only half way each time. But they all add up to 1, or approach 1 as the sequence approaches an infinite number of fractions.

But that's only part of the final answer.

16 meters is the sum of all the downward motions.

You also need to add the upward motions as it bounces back to get the full distances traveled.

4+2+1+1/2+1/4.... are the downward distances, which sum to 8

16+8=24 meters grand total

There's a formula for the sum of an infinite geometric series S=a₁/(1-r) where a₁ is the first term in the series and r is the ratio of the next term in the series a+ar+ar²+ar³+......=S

For the bouncing ball a₁=8 meters for downward motion and a₁=4 for upward motion, and r=1/2 S_downward + S_upward=8/(1-1/2) + 4/(1-1/2)=12/(1/2)=12x2=24 meters