Difference between revisions of "2022 AMC 12B Problems/Problem 23"

Latest revision as of 07:57, 5 December 2022

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-==Problem==
+#REDIRECT [[2022_AMC_10B_Problems/Problem_25]]
-Let <math>x_0,x_1,x_2,\dotsc</math> be a sequence of numbers, where each <math>x_k</math> is either <math>0</math> or <math>1</math>. For each positive integer <math>n</math>, define
-<cmath>S_n = \sum_{k=0}^{n-1} x_k 2^k</cmath>
-Suppose <math>7S_n \equiv 1 \pmod{2^n}</math> for all <math>n \geqslant 1</math>. What is the value of the sum
-<cmath>x_{2019} + 2x_{2020} + 4x_{2021} + 8x_{2022}</cmath>
-<math>\textbf{(A)}~6\qquad\textbf{(B)}~7\qquad\textbf{(C)}~12\qquad\textbf{(D)}~14\qquad\textbf{(E)}~15\qquad</math>
-==Solution==