EXERCISE SET A

1. Suppose that in order to take any step, you need to take a half step before

that. Would you even be able to move? If so, how far? Prove your answers.

2. Suppose three people want to share four identical square-shaped slices of

cheese. They each take a slice, leaving the fourth one, which is then cut into

four identical square-shaped pieces, but a smaller 1/4 slice. They each take

one of these smaller slices, then cut the remaining slice into identical

quarters as before, but even smaller this time, 1/8 slice. And so they

continue until only a tiny crumb is left, which is of no interest and

abandoned. How many slices of cheese, total, would each of them get to eat

if this were to go on indefinitely? Prove your answer in two different ways.

The formula for the sum is S = t1/(1 - r) where t1 is the first term and r is the

ratio (the second term divided by the first term). The three people start with a

whole slice (t1 = 1), then begin cutting slices.

Quartered slices...

Dividing...

In both cases...