Question 106
Suppose that $a_k \gt 0$, $k = 1, 2, \dots, 2008$.
Prove that if and only if $\sum_{k=1}^{2008} a_k \gt 1$, there is a sequence $\{x_n\}$ satisfying
(1) $0 = x_0 \lt x_n \lt x_{n+1}$, $n = 1, 2, 3, \dots$;
(2) $\lim_{n \to \infty} x_n$ exists;
(3) $x_n - x_{n-1} = \sum_{k=1}^{2008} a_k x_{n+k} - \sum_{k=0}^{2007} a_{k+1} x_{n+k}$, $n = 1, 2, 3, \dots$
Prove that if and only if $\sum_{k=1}^{2008} a_k \gt 1$, there is a sequence $\{x_n\}$ satisfying
(1) $0 = x_0 \lt x_n \lt x_{n+1}$, $n = 1, 2, 3, \dots$;
(2) $\lim_{n \to \infty} x_n$ exists;
(3) $x_n - x_{n-1} = \sum_{k=1}^{2008} a_k x_{n+k} - \sum_{k=0}^{2007} a_{k+1} x_{n+k}$, $n = 1, 2, 3, \dots$