Question 8
A sequence $a_{1}, a_{2}, \ldots$ of positive integers satisfies $a_{1}=1$ and
$$
a_{n+1}=2^{a_{n}}+a_{n}
$$
for $n \geq 1$. Prove that $a_{1}, a_{2}, \ldots, a_{243}$ leave distinct remainders when divided by $243$.
$$
a_{n+1}=2^{a_{n}}+a_{n}
$$
for $n \geq 1$. Prove that $a_{1}, a_{2}, \ldots, a_{243}$ leave distinct remainders when divided by $243$.