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AMC (AIMO validation set) / https://artofproblemsolving.com/wiki/index.php/2022_AMC_12B_Problems/Problem_23 / Let x_0,x_1,x_2,dotsc be a sequence of numbers, where each x_(k) is either 0 or 1. For each positive integer n, define S_(n)=∑_(k=0)^(n−1)x_(k)2^(k) Suppose 7S_n equiv 1 pmod{2^n} …

Problem

Answer published by the source. Consult the official source to check your work against its answer.

problem

Let $x_0,x_1,x_2,\dotsc$ be a sequence of numbers, where each

x_k

is either

0

1

. For each positive integer

n

, define

S_n = \sum_{k=0}^{n-1} x_k 2^k

Suppose $7S_n \equiv 1 \pmod{2^n}$ for all

n \geq 1

. What is the value of the sum \[x_{2019} + 2x_{2020} + 4x_{2021} + 8x_{2022}?\]

Plain-text mathematical notation (without MathML)

Let $x_0,x_1,x_2,\dotsc$ be a sequence of numbers, where each x_(k) is either 0 or 1. For each positive integer n, define 
S_(n)=∑_(k=0)^(n−1)x_(k)2^(k)
Suppose $7S_n \equiv 1 \pmod{2^n}$ for all n≥1. What is the value of the sum  
\[x_{2019} + 2x_{2020} + 4x_{2021} + 8x_{2022}?\]

Original LaTeX notation

Let $x_0,x_1,x_2,\dotsc$ 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 $7S_n \equiv 1 \pmod{2^n}$ for all $n \geq 1$. What is the value of the sum  
\[x_{2019} + 2x_{2020} + 4x_{2021} + 8x_{2022}?\]

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Source and history

upstream_id / Original problem ID: https://artofproblemsolving.com/wiki/index.php/2022_AMC_12B_Problems/Problem_23
task_id: f8d20723-8164-535e-b987-b27a5bb3b793
task_key: default--train--https~3a~2f~2fartofproblemsolving~2ecom~2fwiki~2findex~2ephp~2f2022~5fAMC~5f12B~5fProblems~2fProblem~5f23
task_revision_id: 2
config / split: default / train

Official source

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