2013 AMC 12B Problem 7

Below is the professionally curated solution for Problem 7 of the 2013 AMC 12B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2013 AMC 12B solutions, or check the answer key.

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Concepts:triangular number

Difficulty rating: 1380

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?

$2$

$3$

$5$

$6$

$8$

Solution:

After the turn that counts up to $n,$ exactly $1 + 2 + \cdots + n = \tfrac12 n(n+1)$ numbers have been said. For $n = 9$ that is $45.$ The next turn starts $1, 2, \ldots,$ so the $53$ rd number is the $8$ th number of that turn, namely $8.$ Thus, the correct answer is E.

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