Difference between revisions of "2013 AMC 12B Problems/Problem 7"

m
(Problem)
Line 1: Line 1:
 
==Problem==
 
==Problem==
 
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 <math>53^{\text{rd}}</math> number said?<br \>
 
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 <math>53^{\text{rd}}</math> number said?<br \>
 +
 
<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>
 
  
 
==Solution==
 
==Solution==

Revision as of 18:21, 22 February 2013

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 $53^{\text{rd}}$ 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 $\boxed{\textbf{(E) }8}$.

See also

2013 AMC 12B (ProblemsAnswer KeyResources)
Preceded by
Problem 6
Followed by
Problem 8
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AMC 12 Problems and Solutions