AMC12 2019 A

AMC12 2019 A · Q17

AMC12 2019 A · Q17. It mainly tests Polynomials, Sequences & recursion (algebra).

Let $s_k$ denote the sum of the $k$th powers of the roots of the polynomial $x^3 - 5x^2 + 8x - 13$. In particular, $s_0 = 3$, $s_1 = 5$, and $s_2 = 9$. Let $a$, $b$, and $c$ be real numbers such that $s_{k+1} = a s_k + b s_{k-1} + c s_{k-2}$ for $k = 2, 3, \dots$. What is $a + b + c$?

设$s_k$表示多项式$x^3 - 5x^2 + 8x - 13$的根的$k$次幂之和。特别地，$s_0 = 3$，$s_1 = 5$，且$s_2 = 9$。设$a$，$b$，$c$为实数使得$s_{k+1} = a s_k + b s_{k-1} + c s_{k-2}$对$k = 2, 3, \dots$成立。求$a + b + c$？

(A) -6 -6

(B) 0 0

(D) 10 10

(E) 26 26

Answer

Correct choice: (D)

正确答案：（D）

Solution

Answer (D): Let $r$ denote a root of the polynomial. Then $r^3 = 5r^2 - 8r + 13$. For $k \ge 2$ multiplying both sides by $r^{k-2}$ gives $r^{k+1} = 5r^k - 8r^{k-1} + 13r^{k-2}$. Summing this identity over all roots $r$ yields $s_{k+1} = 5s_k - 8s_{k-1} + 13s_{k-2}$. Thus $a + b + c = 5 - 8 + 13 = 10$.

答案（D）：设 $r$ 为该多项式的一个根，则 $r^3 = 5r^2 - 8r + 13$。当 $k \ge 2$ 时，两边同乘 $r^{k-2}$ 得 $r^{k+1} = 5r^k - 8r^{k-1} + 13r^{k-2}$。对所有根 $r$ 将该恒等式求和，得到 $s_{k+1} = 5s_k - 8s_{k-1} + 13s_{k-2}$。因此 $a + b + c = 5 - 8 + 13 = 10$。

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