Algebra homework

Question

Suppose that you drop a ball from a window 50 meters above the ground. The ball bounces 50% of its previous height with each bounce. If a 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 25 bounce?

A. 75m

B. 100m

C. 150m

D. 200m

Answer 1

All distances are in meters.

Down 50, Up 25, Down 25, up 12.5, down 12.5, .....

50 + (25 + 25) + (12.5 + 12.5) + . . .

50 + 50*(1/2)^0, 50*(1/2)^1 + 50*(1/2)^2 + . . . = 50 + 100 = 150 meters

Where did we get the 100 from: The first term of the series is 50, a. r = 1/2; The sum of an infinite number of terms converges to a/(1-r) = 50/(1-(1/2)) = 50/(1/2) = 100, as long as |r| <1.

Answer 2

you can look at the problem as a geometric sequence

50+25+12.5+...

S_n=a₁(1-rⁿ)/(1-r)

S_n=50(1-0.5²⁵)/(1-0.5)

S_n=50(1-.0000000298)/0.5

S_n=100(0.9999999702)

S_n=99.99999702

S_n=100 (basically because 0.25²⁵ is an approximate number

now all the numbers are doubled except 50 which appears only once

therefore, double 100 and subtract the extra 50

2*100-50=200-50=150 meters is the sum