Problem 230 (difficulty: 4/10)

Give examples such that \(\displaystyle a_n \to \infty\) and

 (1)  \(\displaystyle a_{2n} -a_n \to 0;\)      (2)  \(\displaystyle a_{n^2} -a_n \to 0;\)      (3)  \(\displaystyle a_{2^n} -a_n \to 0;\)

Give me another random problem!