AMC12 2009 A

AMC12 2009 A · Q7

AMC12 2009 A · Q7. It mainly tests Arithmetic sequences basics.

The first three terms of an arithmetic sequence are $2x - 3$, $5x - 11$, and $3x + 1$ respectively. The $n$th term of the sequence is $2009$. What is $n$?

一个等差数列的前三项分别是 $2x - 3$、$5x - 11$ 和 $3x + 1$。该数列的第 $n$ 项是 $2009$。$n$ 是多少？

(A) 255 255

(B) 502 502

(D) 1506 1506

(E) 8037 8037

Answer

Correct choice: (B)

正确答案：（B）

Solution

As this is an arithmetic sequence, the difference must be constant: $(5x-11) - (2x-3) = (3x+1) - (5x-11)$. This solves to $x=4$. The first three terms then are $5$, $9$, and $13$. In general, the $n$th term is $1+4n$. Solving $1+4n=2009$, we get $n=\boxed{502}$.

因为这是等差数列，公差必须为常数：$(5x-11) - (2x-3) = (3x+1) - (5x-11)$。解得 $x=4$。此时前三项为 $5$、$9$、$13$。一般地，第 $n$ 项为 $1+4n$。解 $1+4n=2009$，得 $n=\boxed{502}$。

Topics

AR.012 Arithmetic sequences basics