MathDB

A large pond contains infinitely many lily pads labelled

1

2

3

...

, placed in a line, where for each

k

, lily pad

k + 1

is one unit to the right of lily pad

k

. A frog starts at lily pad

100

. Each minute, if the frog is at lily pad

n

, it hops to lily pad

n + 1

with probability

\frac{n-1}{n}

, and hops all the way back to lily pad

1

with probability

\frac{1}{n}

. Let

N

be the position of the frog after

1000

minutes. What is the expected value of

N

Problem Statement