2013 AMC 12B Problem 7

Attempt Problem 7 of the 2013 AMC 12B below, then check your answer against the professionally curated solution 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.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

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

The turn that ends at

n

consists of someone saying

1, 2, \ldots, n

After the turn ending at

n,

a total of

\tfrac12 n(n+1)

numbers have been said

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.

Problem 7 in other years