MathDB

Let

r

be a positive integer, and let

a_0 , a_1 , \cdots

be an infinite sequence of real numbers. Assume that for all nonnegative integers

m

and

s

there exists a positive integer

n \in [m+1, m+r]

such that

a_m + a_{m+1} +\cdots +a_{m+s} = a_n + a_{n+1} +\cdots +a_{n+s}

Prove that the sequence is periodic, i.e. there exists some

p \ge 1

such that

a_{n+p} =a_n

for all

n \ge 0

Problem Statement