2019 AMC 12A Problem 17

Below is the professionally curated solution for Problem 17 of the 2019 AMC 12A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2019 AMC 12A solutions, or check the answer key.

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Concepts:Newton’s Sums polynomial recursion

Difficulty rating: 1860

17.

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 $k = 2, 3, \ldots.$ What is $a + b + c?$

$-6$

$0$

$6$

$10$

$26$

Solution:

Every root $r$ satisfies $r^3 = 5r^2 - 8r + 13,$ so $r^{k+1} = 5r^k - 8r^{k-1} + 13r^{k-2}.$

Summing over the three roots gives $s_{k+1} = 5s_k - 8s_{k-1} + 13s_{k-2},$ so $a = 5,$ $b = -8,$ $c = 13.$

Therefore $a + b + c = 5 - 8 + 13 = 10.$

Thus, the correct answer is D.

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