2016 AIME I Problems/Problem 13
Notice that we don't really care about what the -coordinate of the frog is. So let's let denote the expected number of times Freddy will jump at a coordinate of until he reaches the line . So therefore we want to find .
So we have . Suppose Freddy is at . He has a probability of moving horizontally, chance of moving up and chance of moving down. So we have So we get the recursion . Rearranging we see . That means the difference between consecutive terms goes down by each time. So for convenience let's let and . So that means Yes, this is a quadratic. Now, notice that since there is a boundary, we have to give special care to . We have so this turns into and this results in . So now we know Now, we also have that so that gives us so . So now we know so plugging in we get