Counting Two Ways

An infinite sequence $P \colon a_1, a_2, \ldots$ satisfies $a_i \in \mathbb{N}^*$ and $a_i \leqslant a_{i+1}$. For $k \in \mathbb{N}^*$, let $T_k(P)$ be the smallest element of $\{n \mid a_n \geqslant k\}$.

1. For the sequence $P \colon 1, 3, 4, 7, \ldots$, write down $T_1(P), \ldots, T_5(P)$. 2. If $T_k(P) = 2k - 1$ for all $k$, find the sum of the first $n$ terms of $P$. 3. Given $a_{20} = 46$, find