Algebra 2 sequences

Question

A ball is dropped from a height of 40 feet. Each time it hits the ground, it bounces three-fourths of its previous height. Find the total distance the ball has traveled before coming to rest.

Answer 1

Ignore the first drop of 40 feet for now.

The next height will be (3/4)40 = 30. All the heights will be an infinite geometric series. The travel distance of one bounce is twice the height to account for rising and falling.

2(30 + 30(3/4) + 30(3/4)²+ ....)

Let g₁ = 30 and r = 3/4.

S_n = 2∑g₁r^n-1 = 2g₁(1/(1-r)) for n = 1 to ∞

Total distance traveled is the original height plus the geometric series:

distance = 40 + 2(30)(1/(1-3/4))

40+60(4) = 280 feet.

Another way to look at this is to add the down distance and the up distances.

Down distance = 40(1/(1-3/4)) = 160 feet

Up distance = 30(1/(1-3/4) = 120 feet

Up + Down = 280 feet