MathNet 2017-part-03 — 176 problems (EN–VI) Question 53

The sequence $\{x_n\}$ is defined by $x_1 = 5$ and $x_{k+1} = x_k^2 - 3x_k + 3$ for $k = 1, 2, 3, \dots$. Prove that $x_k \gt 3^{2^{k-1}}$ for any positive integer $k$.