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Source: 2022 AMC 10B #25 / 2022 AMC 12B #23

November 17, 2022
AMCAMC 10AMC 122022 AMC2022 AMC 10B2022 AMC 12BSequence

Problem Statement

Let x0x_{0}, x1x_{1}, x2x_{2}, ⋯\cdots be a sequence of numbers, where each xkx_{k} is either 00 or 11. For each positive integer nn, define Sn=∑k=0n−1xk2kS_{n} = \displaystyle\sum^{n-1}_{k=0}{x_{k}2^{k}}
Suppose 7Sn≡1(mod2n)7S_{n} \equiv 1\pmod {2^{n}} for all n≥1n\geq 1. What is the value of the sum x2019+2x2020+4x2021+8x2022?x_{2019}+2x_{2020}+4x_{2021}+8x_{2022}?
<spanclass=′latex−bold′>(A)</span> 6<spanclass=′latex−bold′>(B)</span> 7<spanclass=′latex−bold′>(C)</span> 12<spanclass=′latex−bold′>(D)</span> 14<spanclass=′latex−bold′>(E)</span> 15 <span class='latex-bold'>(A)</span>\ 6 \qquad <span class='latex-bold'>(B)</span>\ 7 \qquad <span class='latex-bold'>(C)</span>\ 12 \qquad <span class='latex-bold'>(D)</span>\ 14 \qquad <span class='latex-bold'>(E)</span>\ 15
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