Question 23
Let $x_0,$ $x_1,$ $x_2,$ $\ldots$ be a sequence of numbers, where each $x_k$ is either $0$ or $1.$ For each positive integer $n,$ define $$
S_n = \sum_{k=0}^{n-1} x_k 2^k.
$$ Suppose $7 S_n \equiv 1 \pmod{2^n}$ for all $n \ge 1.$ What is the value of the sum $$
x_{2019} + 2x_{2020} + 4x_{2021} + 8x_{2022}?
$$
S_n = \sum_{k=0}^{n-1} x_k 2^k.
$$ Suppose $7 S_n \equiv 1 \pmod{2^n}$ for all $n \ge 1.$ What is the value of the sum $$
x_{2019} + 2x_{2020} + 4x_{2021} + 8x_{2022}?
$$