Sums Within the Set

A set of numbers $A = \{a_1, a_2, \dots, a_n\}$ with $1 = a_1 < a_2 < \cdots < a_n$ and $n \geqslant 2$ has property $P$ if for every $k$ with $2 \leqslant k \leqslant n$ there exist $i, j$ with $1 \leqslant i \leqslant j \leqslant n$ such that $a_k = a_i + a_j$.

1. Determine whether $\{1, 3, 5\}$ and $\{1, 2, 3, 6\}$ have property $P$, with justification. 2. Let $S_n = a_1 + a_2 + \cdots + a_n$. Prove that $2a_n - 1 \leqslant S_n$. 3. If $a_n = 36$, find the minimum possible sum of all elements of $A$.