AMC10 2008 B

AMC10 2008 B · Q11

AMC10 2008 B · Q11. It mainly tests Sequences & recursion (algebra).

Suppose that $(u_n)$ is a sequence of real numbers satisfying $u_{n+2} = 2u_{n+1} + u_n$, and that $u_3 = 9$ and $u_6 = 128$. What is $u_5$?

假设 $(u_n)$ 是一个实数序列，满足 $u_{n+2} = 2u_{n+1} + u_n$，且 $u_3 = 9$，$u_6 = 128$。求 $u_5$ 的值。

(A) 40 40

(B) 53 53

(D) 88 88

(E) 104 104

Answer

Correct choice: (B)

正确答案：（B）

Solution

Answer (B): Note that $u_5 = 2u_4 + 9$ and $128 = u_6 = 2u_5 + u_4 = 5u_4 + 18$. Thus $u_4 = 22$, and it follows that $u_5 = 2 \cdot 22 + 9 = 53$.

答案（B）：注意 $u_5 = 2u_4 + 9$ 且 $128 = u_6 = 2u_5 + u_4 = 5u_4 + 18$。因此 $u_4 = 22$，从而 $u_5 = 2 \cdot 22 + 9 = 53$。

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