Question 31
Let $\alpha$ be a real number larger than $1$. Define a sequence $(s_n)_{n \ge 1}$ as follows: $s_1 = 1$, $s_2 = \alpha$; if $s_1, s_2, s_3, \dots, s_{2^n}$ are defined for some $n \ge 1$, then $s_{2^n+1}, s_{2^n+2}, \dots, s_{2^{n+1}}$ are defined by $s_j = \alpha s_{j-2^n}$, for $2^n+1 \le j \le 2^{n+1}$. (Thus the first few terms are $1, \alpha, \alpha, \alpha^2, \alpha, \alpha^2, \alpha^2, \alpha^3, \dots$) Let $c_n = s_1+s_2+s_3+\dots+s_n$. If $n = 2^{e_0} + 2^{e_1} + \dots + 2^{e_k}$, where $e_0 \gt e_1 \gt \dots \gt e_k \ge 0$ is the binary representation of a positive integer $n$, prove that
$$
c_n = (1 + \alpha)^{e_0} + \alpha(1 + \alpha)^{e_1} + \alpha^2(1 + \alpha)^{e_2} + \dots + \alpha^k(1 + \alpha)^{e_k}.
$$
$$
c_n = (1 + \alpha)^{e_0} + \alpha(1 + \alpha)^{e_1} + \alpha^2(1 + \alpha)^{e_2} + \dots + \alpha^k(1 + \alpha)^{e_k}.
$$