Question 91
Given positive integers $u, v$, the sequence $\{a_n\}$ is defined as: $a_1 = u + v$, and for $m \ge 1$,
$$
\begin{cases} a_{2m} = a_m + u, \\ a_{2m+1} = a_m + v. \end{cases}
$$
Denote $S_m = a_1 + a_2 + \dots + a_m$ ($m = 1, 2, \dots$). Prove that there are infinite terms in sequence $\{S_n\}$ that are square numbers.
$$
\begin{cases} a_{2m} = a_m + u, \\ a_{2m+1} = a_m + v. \end{cases}
$$
Denote $S_m = a_1 + a_2 + \dots + a_m$ ($m = 1, 2, \dots$). Prove that there are infinite terms in sequence $\{S_n\}$ that are square numbers.