MathDB

Patty is standing on a line of planks playing a game. Define a block to be a sequence of adjacent planks, such that both ends are not adjacent to any planks. Every minute, a plank chosen uniformly at random from the block that Patty is standing on disappears, and if Patty is standing on the plank, the game is over. Otherwise, Patty moves to a plank chosen uniformly at random within the block she is in; note that she could end up at the same plank from which she started. If the line of planks begins with

n

planks, then for sufficiently large n, the expected number of minutes Patty lasts until the game ends (where the first plank disappears a minute after the game starts) can be written as

P(1/n)f(n) + Q(1/n)

, where

P,Q

are polynomials and

f(n) =\sum^n_{i=1}\frac{1}{i}

. Find

P(2023) + Q(2023)

Problem Statement