AMC10 2011 A

AMC10 2011 A · Q17

AMC10 2011 A · Q17. It mainly tests Sequences & recursion (algebra).

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 $A + H$?

在一个八项数列 $A, B, C, D, E, F, G, H$ 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

Answer (C): Note that for any four consecutive terms, the first and last terms must be equal. For example, consider $B, C, D,$ and $E$; because $$B + C + D = 30 = C + D + E,$$ we must have $B = E$. Hence $A = D = G$, and $C = F = 5$. The required sum $A + H = G + (30 - G - F) = 30 - 5 = 25$. OR Note that $$\begin{array}{rcl} A + C + H &=& (A + B + C) - (B + C + D) + (C + D + E) \\ && - (E + F + G) + (F + G + H) \\ &=& 3\cdot 30 - 2\cdot 30 = 30. \end{array}$$ Hence $A + H = 30 - C = 25$.

答案（C）：注意任意四个连续项中，第一项和最后一项必须相等。比如考虑 $B, C, D, E$；因为 $$B + C + D = 30 = C + D + E,$$ 所以必须有 $B = E$。因此 $A = D = G$，且 $C = F = 5$。所求和为 $A + H = G + (30 - G - F) = 30 - 5 = 25$。或者注意 $$\begin{array}{rcl} A + C + H &=& (A + B + C) - (B + C + D) + (C + D + E) \\ && - (E + F + G) + (F + G + H) \\ &=& 3\cdot 30 - 2\cdot 30 = 30. \end{array}$$ 因此 $A + H = 30 - C = 25$。

Topics

AL.012 Sequences & recursion (algebra)