MathNet 2017-part-04 — 176 problems (EN–VI) Question 136

Consider the sequence of rational numbers defined by $x_1 = 4/3$, and $x_{n+1} = x_n^2/(x_n^2 - x_n + 1)$, $n \ge 1$. Show that the numerator of the lowest term expression of each sum $\sum_{k=1}^n x_k$ is a perfect square.