Question 20
Let m be a fixed integer greater than 1. The sequence $ x_{0}, x_{1}, x_{2}, \ldots $ is defined as follows:
$$ x_{i}=\begin{cases}2^{i},&\text{if }0\leq i\leq m-1;\\\sum_{j=1}^{m}x_{i-j},&\text{if } i\geq m.\end{cases} $$
Find the greatest k for which the sequence contains k consecutive terms divisible by m.
$$ x_{i}=\begin{cases}2^{i},&\text{if }0\leq i\leq m-1;\\\sum_{j=1}^{m}x_{i-j},&\text{if } i\geq m.\end{cases} $$
Find the greatest k for which the sequence contains k consecutive terms divisible by m.