Question 31
The equation $x^3 + 2x - 1 = 0$ has only one real root $r$. Note that $0.4 \lt r \lt 0.5$.
a. Prove that $\sum_{k=0}^{\infty} r^{3k+1} = \frac{1}{2}$.
b. Suppose that $(a_k)$ be a strictly increasing sequence of positive integral numbers such that $\sum_{k=0}^{\infty} r^{a_k} = \frac{1}{2}$. Prove that $a_k = k+1$ for all $k = 0, 1, 2, \ldots$
a. Prove that $\sum_{k=0}^{\infty} r^{3k+1} = \frac{1}{2}$.
b. Suppose that $(a_k)$ be a strictly increasing sequence of positive integral numbers such that $\sum_{k=0}^{\infty} r^{a_k} = \frac{1}{2}$. Prove that $a_k = k+1$ for all $k = 0, 1, 2, \ldots$