Question

A ball is dropped from a height of 10 feet and returns to a height that is one-half of the height from which it fell. The ball continues to bounce half the height of the previous bounce each time. How far will the ball have traveled when it hits the ground for the fourth time?

Answer 1

Ok, so this is a sum of aruthmetic sequence and also use some logic

it goes 10 feet down, then 5 up, 5 down, etc (see attachment)

so the total distance is the sum of all the numbers in the sequence doubled, minus the first term (since it didn't bounce 10 feet up, it started from top)

the sum of an aruthmetic is
Sn=$\frac{a_{1}(1-r^n)}{1-r}$
where a1=first term, r=common ratio and n=which time

first term=10
common ratio is 0.5
n=4

so
2[$\frac{10(1-0.5^4)}{1-0.5}$]-10=answer27.5

the answer is 27.5 feet