Given $f(n) \le 1+f(\lfloor(2n/3)\rfloor)$. We need to find the smallest integer k such that $n \le \lceil(3/2)^k\rceil$ and give an upper bound for $f(\lceil(3/2)^k\rceil)$.
I already know that $\lfloor2/3\lceil(2n/3)^i\rceil\rfloor \le \lceil(3/2)^{i-1}\rceil$ and have the inequality that $f(\lceil(3/2)^k\rceil) \le 1+\lceil(3/2)^{k-1}\rceil$. I have no idea how to process.