AMC10 2013 B

AMC10 2013 B · Q13

AMC10 2013 B · Q13. It mainly tests Arithmetic sequences basics, Basic counting (rules of product/sum).

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 53rd number said?

Jo 和 Blair 轮流从 1 数到对方上次说的数加一。Jo 先说“1”，Blair 接着说“1, 2”。Jo 然后说“1, 2, 3”，依此类推。第 53 个说的数字是多少？

(A) 2 2

(B) 3 3

(D) 6 6

(E) 8 8

Answer

Correct choice: (E)

正确答案：（E）

Solution

Answer (E): Note that Jo starts by saying 1 number, and this is followed by Blair saying 2 numbers, then Jo saying 3 numbers, and so on. After someone completes her turn after saying the number $n$, then $1+2+3+\cdots+n=\frac{1}{2}n(n+1)$ numbers have been said. If $n=9$ then 45 numbers have been said. Therefore there are $53-45=8$ more numbers that need to be said. The $53^{\mathrm{rd}}$ number said is 8.

答案（E）：注意，Jo 先说 1 个数，接着 Blair 说 2 个数，然后 Jo 说 3 个数，依此类推。某人在说完数字 $n$ 后完成她的回合时，那么已经说过的数字个数为 $1+2+3+\cdots+n=\frac{1}{2}n(n+1)$。若 $n=9$，则已经说了 45 个数。因此还需要再说 $53-45=8$ 个数。所说的第 $53^{\mathrm{rd}}$ 个数是 8。

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