{"status": "success", "data": {"description_md": "The silver ratio $\\varepsilon$ is defined as $\\varepsilon=\\sqrt{2}+1$. Consider the sequence $$a_n=\\left \\lfloor \\varepsilon^n-\\dfrac{1}{(-\\varepsilon)^n} \\right \\rfloor $$ for all positive integers $n$. What is the sum of the first $3$ terms in the sequence that are prime?    ", "description_html": "<p>The silver ratio <span class=\"katex--inline\">\\varepsilon</span> is defined as <span class=\"katex--inline\">\\varepsilon=\\sqrt{2}+1</span>. Consider the sequence <span class=\"katex--display\">a_n=\\left \\lfloor \\varepsilon^n-\\dfrac{1}{(-\\varepsilon)^n} \\right \\rfloor</span> for all positive integers <span class=\"katex--inline\">n</span>. What is the sum of the first <span class=\"katex--inline\">3</span> terms in the sequence that are prime?</p>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 5, "problem_name": "TxO Math Bowl 2024 - Individuals B - Problem 11", "can_next": true, "can_prev": true, "nxt": "/problem/txo2024indivsB-p12", "prev": "/problem/txo2024indivsB-p10"}}