A Functional Sum Identity

A sequence $\{a_n\}$ of positive terms has partial sums $S_n$, and for all $m, n \in \mathbb{N}^*$,

1. Find $\dfrac{a_2}{a_1}$. 2. Prove that $\{a_n\}$ is geometric. 3. Sequences $\{c_n\}, \{d_n\}$ satisfy $|c_n| = |d_n| = a_n$. For a given integer $p \ge 3$, their $p$-th partial sums $T_p, R_p$ are equal. Prove that $c_k = d_k$ for every $1 \le k \le p$.