2013 AMC 12B Problem 7

Problem 7 of 25EasierAlgebra

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?

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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.