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 number is spoken in position $53$ ?

$2$

$3$

$5$

$6$

$8$

Answer: E

Concepts:triangular number pattern recognition

Difficulty rating: 1140

Small Hint:

Find the triangular number just before $53$

Big Hint:

The next spoken list starts again at $1$

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

Small Hint:

Big Hint:

Solution:

Problem 13 in Other Years