{"status": "success", "data": {"description_md": "Let $s_k$ denote the sum of the $\\textit{k}$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 $k = 2$, $3$, $....$ What is $a+b+c$?\n\n$\\textbf{(A)} \\; -6 \\qquad \\textbf{(B)} \\; 0 \\qquad \\textbf{(C)} \\; 6 \\qquad \\textbf{(D)} \\; 10 \\qquad \\textbf{(E)} \\; 26$\n___\nFull credit goes to [MAA](https://maa.org/) for authoring these problems. These problems were taken on the [AOPS](https://artofproblemsolving.com/) website.", "description_html": "<p>Let  <span class=\"katex--inline\">s_k</span>  denote the sum of the  <span class=\"katex--inline\">\\textit{k}</span> th powers of the roots of the polynomial  <span class=\"katex--inline\">x^3-5x^2+8x-13</span> . In particular,  <span class=\"katex--inline\">s_0=3</span> ,  <span class=\"katex--inline\">s_1=5</span> , and  <span class=\"katex--inline\">s_2=9</span> . Let  <span class=\"katex--inline\">a</span> ,  <span class=\"katex--inline\">b</span> , and  <span class=\"katex--inline\">c</span>  be real numbers such that  <span class=\"katex--inline\">s_{k+1} = a \\, s_k + b \\, s_{k-1} + c \\, s_{k-2}</span>  for  <span class=\"katex--inline\">k = 2</span> ,  <span class=\"katex--inline\">3</span> ,  <span class=\"katex--inline\">....</span>  What is  <span class=\"katex--inline\">a+b+c</span> ?</p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A)} \\; -6 \\qquad \\textbf{(B)} \\; 0 \\qquad \\textbf{(C)} \\; 6 \\qquad \\textbf{(D)} \\; 10 \\qquad \\textbf{(E)} \\; 26</span> </p>&#10;<hr><p>Full credit goes to <a href=\"https://maa.org/\">MAA</a> for authoring these problems. These problems were taken on the <a href=\"https://artofproblemsolving.com/\">AOPS</a> website.</p>", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 3, "problem_name": "2019 AMC 12A Problem 17", "can_next": true, "can_prev": true, "nxt": "/problem/19_amc12A_p18", "prev": "/problem/19_amc12A_p16"}}