AMC12 2011 A

AMC12 2011 A · Q8

AMC12 2011 A · Q8. It mainly tests Sequences & recursion (algebra), Invariants.

In the eight term sequence $A$, $B$, $C$, $D$, $E$, $F$, $G$, $H$, the value of $C$ is $5$ and the sum of any three consecutive terms is $30$. What is $A+H$?

在八项数列$A$, $B$, $C$, $D$, $E$, $F$, $G$, $H$中，$C$的值为$5$，且任意三个连续项的和为$30$。求$A+H$。

(A) 17 17

(B) 18 18

(D) 26 26

(E) 43 43

Answer

Correct choice: (C)

正确答案：（C）

Solution

Let $A=x$. Then from $A+B+C=30$, we find that $B=25-x$. From $B+C+D=30$, we then get that $D=x$. Continuing this pattern, we find $E=25-x$, $F=5$, $G=x$, and finally $H=25-x$. So $A+H=x+25-x=25 \rightarrow \boxed{\textbf{C}}$

设$A=x$。由$A+B+C=30$得$B=25-x$。由$B+C+D=30$得$D=x$。继续此规律可得$E=25-x$，$F=5$，$G=x$，最后$H=25-x$。因此$A+H=x+25-x=25 \rightarrow \boxed{\textbf{C}}$

Topics