Difference between revisions of "2013 AMC 10B Problems/Problem 13"
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| + | ==Problem== | ||
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Jo and Blair take turns counting from <math>1</math> to one more than the last number said by the other person. Jo starts by saying "<math>1</math>", so Blair follows by saying "<math>1, 2</math>" . Jo then says "<math>1, 2, 3</math>" , and so on. What is the 53rd number said? | Jo and Blair take turns counting from <math>1</math> to one more than the last number said by the other person. Jo starts by saying "<math>1</math>", so Blair follows by saying "<math>1, 2</math>" . Jo then says "<math>1, 2, 3</math>" , and so on. What is the 53rd number said? | ||
<math> \textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ 8 </math> | <math> \textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ 8 </math> | ||
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| + | ==Solution== | ||
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| + | 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," <math>1+2+3+4+5+6+7+8+9=45</math> numbers have been said. In the tenth turn, the eighth number will be the 53rd number said, as <math>53-45=8</math>. Since we're starting from 1 each time, the 53rd number said will be <math>\boxed{\textbf{(E) }8}</math>. | ||
Revision as of 19:55, 21 February 2013
Problem
Jo and Blair take turns counting from 
to one more than the last number said by the other person. Jo starts by saying "", so Blair follows by saying "
" . Jo then says "
" , and so on. What is the 53rd number said?
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," 
numbers have been said. In the tenth turn, the eighth number will be the 53rd number said, as 
. Since we're starting from 1 each time, the 53rd number said will be 
.
