Question 73
For sets $S$ and $T$ consisting of positive real numbers, define $S + T = \{s + t \mid s \in S, t \in T\}$ and $\frac{1}{S} = \{\frac{1}{s} \mid s \in S\}$. Define the sets $A_1, A_2, A_3, \dots$ recursively by $A_1 = \{1\}$ and
$$
A_n = \bigcup_{i=1}^{n-1} \left( (A_i + A_{n-i}) \cup \left( \frac{1}{\frac{1}{A_i} + \frac{1}{A_{n-i}}} \right) \right)
$$
for all integers $n \ge 2$. Prove that for all integers $n \ge 1$ we have
$$
2^{n-1} \le |A_n| \le 8^n.
$$
$$
A_n = \bigcup_{i=1}^{n-1} \left( (A_i + A_{n-i}) \cup \left( \frac{1}{\frac{1}{A_i} + \frac{1}{A_{n-i}}} \right) \right)
$$
for all integers $n \ge 2$. Prove that for all integers $n \ge 1$ we have
$$
2^{n-1} \le |A_n| \le 8^n.
$$