Convergence of geometric series factor 3/4 word problem

Question

I came across this problem on AoPS: A rubber ball is dropped from a 100 ft tall building. Each time it bounces, it rises to three quarters its previous height. So, after its first bounce it rises to 75 ft, and after its second bounce it rises to 3/4 of 75 ft, and so on forever. What is the total distance the ball travels? I reduced it to a fairly simple geometric series: $$ \sum\limits_{n = 0}^\infty {100(\frac{3}{4})^{n}} $$ Using $ \sum\limits_{n = 0}^\infty {ax^{n} = \frac{a}{{1-x}}} $ with a=100 and x=3/4, the answer I got was 400 but the answer key said the answer was 700. Did I read the problem wrong and the distance was actually supposed to be measured differently? I don't think the answer key is wrong since the rest of its answers checked out. Any help would be appreciated. Thx in advance. P.S. Not really sure what tags to put on this so pls add a tag that will fit the question.

Answer 1

You counted only the distance the ball traveled downward on each bounce. The answer key is counting the distance the ball traveled in both directions, up and down.

So, $400$ feet downward and $300$ feet upward adds up to $700$ feet.

Answer 2

If you ignore the first drop and the first rise, i.e. the first $175$ feet, what remains is three quarters of the total.

$$t-175=\frac34t\implies t=700.$$