Question 3
For real numbers $a_1,\ldots,a_k,b_1,\ldots,b_k$, define
$$X_n=\sum_{i=1}^k\lfloor a_i n+b_i\rfloor\qquad(n=1,2,\ldots).$$
If $(X_n)$ is an arithmetic progression, prove that $\sum_{i=1}^k a_i$ is an integer. Here $\lfloor r\rfloor$ is the greatest integer not exceeding $r$.
$$X_n=\sum_{i=1}^k\lfloor a_i n+b_i\rfloor\qquad(n=1,2,\ldots).$$
If $(X_n)$ is an arithmetic progression, prove that $\sum_{i=1}^k a_i$ is an integer. Here $\lfloor r\rfloor$ is the greatest integer not exceeding $r$.