2022 AMC 10B Problems/Problem 25
Problem
Let be a sequence of numbers, where each
is either 0 or 1. For each positive
integer
, define
Suppose for all
.
What is the value of the sum
Solution (Base-2 Analysis)
We solve this problem with base 2.
To avoid any confusion, for a base-2 number, we index the th rightmost digit as digit
.
We have .
In the base-2 representation, is equivalent to
In the rest of the analysis, to lighten notation, we ease the base-2 subscription from all numbers. The equation above can be reformulated as:
$$ (Error compiling LaTeX. Unknown error_msg) \begin{tabular}{ccccccccc}
&& 0 &
& 0 & 0 & 0 & 0 & 0 \\ & & & & & & & & 1 \\
& &
&
&
&
&
&
&
\\ \hline %or \bottomrule if using the `booktabs` package &
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&
&
&
&
& 0 & 0 & 0\\ \end{tabular}
$$ (Error compiling LaTeX. Unknown error_msg)
Therefore, ,
, and for
,
.
Therefore,
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
Video Solution
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)