Question 74
Define the Fibonacci numbers $F_n$ as follows: $F_0 = 0, F_1 = 1, F_2 = 1$, and $F_{n+2} = F_{n+1} + F_n$ for all positive integers $n$.
Prove that there is a positive integer $N$ with the following property: among the terms of any non-constant arithmetic progression of positive integers of length $N$, there must be a term which, when expressed as a sum of Fibonacci numbers, requires at least 2024 of them ($F_i$'s can be repeated).
Prove that there is a positive integer $N$ with the following property: among the terms of any non-constant arithmetic progression of positive integers of length $N$, there must be a term which, when expressed as a sum of Fibonacci numbers, requires at least 2024 of them ($F_i$'s can be repeated).