MathNet 2010 — Part 02 Question 162

Prove that equation $2x^3 + 5x - 2 = 0$ has exactly one real root (denoted as $r$), and there is a unique strictly increasing sequence $\{a_n\}$ such that $\frac{2}{5} = r^{a_1} + r^{a_2} + r^{a_3} + \dots$.