IMC 2003 · Day 1 · P1

(a) Let $b_n = \dfrac{a_n}{\left( \frac{3}{2} \right)^{n-1}}$. Then $a_{n+1} > \frac{3}{2} a_n$ is equivalent to $b_{n+1} > b_n$, thus the sequence $(b_n)$ is strictly increasing. Each increasing sequence has a finite limit or tends to infinity.

(b) For all $\alpha > 1$ there exists a sequence $1 = b_1 < b_2 < \dots$ which converges to $\alpha$. Choosing $a_n = \left( \frac{3}{2} \right)^{n-1} b_n$, we obtain the required sequence $(a_n)$.