A Grid of Progressions

A sequence $\{a_n\}$ has exactly $m^2$ terms (with $m > 2$). The first block $a_1, a_2, \dots, a_m$ is an arithmetic sequence with nonzero common difference, and for each $i \in \{1, 2, \dots, m\}$, the subsequence $a_i, a_{i+m}, \dots, a_{i+(m-1)m}$ is arithmetic, with the same nonzero common difference for every $i$.

1. If $m = 3$, $a_1 = 1$, $a_4 = 3$, $a_9 = 9$, find the sum of all terms of $\{a_n\}$. 2. Prove that $a_1, a_{(m+1)+1}, a_{2(m+1)+1}, \dots, a_{(m-1)(m+1)+1}$ form an arithmetic sequence. 3. Choose three numbers $p < q < r$ at random from $1, 2, \dots, m^2$, and let $P_m$ be the probability that $p, q, r$ form an arithmetic sequence and $a_p, a_q, a_r$ also form an arithmetic sequence. Prove that