Question

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?

$2$

$3$

$5$

$6$

$8$

Answer: E

Concepts:triangular number pattern recognition

Difficulty rating: 1140

Solution:

Through the turn ending with $n$ , the total number of numbers spoken is $1+2+\cdots+n=\frac{n(n+1)}2.$

After the turn ending with $9$ , exactly $\frac{9\cdot10}{2}=45$ numbers have been spoken. The next turn is the list $1,2,\ldots,10$ , so its eighth entry is the $53$ rd number spoken. That entry is $8$ .

Thus, the correct answer is E .

2013 AMC 10B Problem 13

Answer: E

Difficulty rating: 1140

Solution:

Problem 13 in Other Years