Question 70
1. For integer $n \geq 3$, suppose that $0 \lt a_{1} \lt a_{2} \lt \ldots \lt a_{n}$ is an arithmetic sequence and $0 \lt b_{1} \lt b_{2} \lt \ldots \lt b_{n}$ is a geometric sequence with $a_{1} = b_{1}$, $a_{n} = b_{n}$. Prove that $a_{k} \gt b_{k}$ for all $k = 2, 3, \ldots, n-1$.
2. Prove that for every positive integer $n \geq 3$, there exist an integer arithmetic sequence ($a_{n}$) and an integer geometric sequence ($b_{n}$) such that
$$
0 \lt b_{1} \lt a_{1} \lt b_{2} \lt a_{2} \lt \cdots \lt b_{n} \lt a_{n}
$$
2. Prove that for every positive integer $n \geq 3$, there exist an integer arithmetic sequence ($a_{n}$) and an integer geometric sequence ($b_{n}$) such that
$$
0 \lt b_{1} \lt a_{1} \lt b_{2} \lt a_{2} \lt \cdots \lt b_{n} \lt a_{n}
$$