2013 AMC 10B Problems/Problem 13

Problem

Jo and Blair take turns counting from $1$ to one more than the last number said by the other person. Jo starts by saying " $1$ ", so Blair follows by saying " $1, 2$ " . Jo then says " $1, 2, 3$ " , and so on. What is the $53^{rd}$ number said?

$\textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ 8$

Solution

We notice that the number of numbers said is incremented by one each time; that is, Jo says one number, then Blair says two numbers, then Jo says three numbers, etc. Thus, after nine "turns," $1+2+3+4+5+6+7+8+9=45$ numbers have been said. In the tenth turn, the eighth number will be the 53rd number said, as $53-45=8$ . Since we're starting from 1 each time, the 53rd number said will be $\boxed{\textbf{(E) }8}$ .

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2013 AMC 10B Problems/Problem 13

Problem

Solution