{"kind":"task","effective_mode":"full","benchmark":{"kind":"benchmark","effective_mode":"full","slug":"math","formal_name":"MATH","introduction":"MATH organises 12,500 high-school competition problems into seven subjects and five difficulty levels, each with a full worked solution. It underlies much of the later mathematics evaluation work; this catalogue imports the full test split: 5,000 problems across all seven subjects.","introduction_ja":"","introduction_en":"","category":"Category not supplied","task_count":null,"acquisition_status":"Acquisition status not supplied","official_url":"https://github.com/hendrycks/math","indexing_mode":"index","profile":{"resources":[],"task_format":"","scoring":"","metric":"","size":"","answer_access":"","license":"","citation":"","maintainer":"","released":"","why_hard":"","related":[],"history":"","construction":"","limitations":"","saturation":"","scores":[]}},"task_id":"1176352c-1064-5a1c-95a9-2659d887243f","task_key":"precalculus--test--1176352c-1064-5a1c-95a9-2659d887243f","task_revision_id":"1","upstream_id":"","short_description":"Let $M_n$ be the $n \\times n$ matrix with entries as follows: for $1 \\le i \\le…","config":"precalculus","split":"test","body":"{\"level\":\"Level 5\",\"problem\":\"Let $M_n$ be the $n \\\\times n$ matrix with entries as follows: for $1 \\\\le i \\\\le n$, $m_{i,i} = 10$; for $1 \\\\le i \\\\le n - 1$, $m_{i+1,i} = m_{i,i+1} = 3$; all other entries in $M_n$ are zero. Let $D_n$ be the determinant of matrix $M_n$. Find\\n\\\\[\\\\sum_{n=1}^{\\\\infty} \\\\frac{1}{8D_n+1}.\\\\]Note: The determinant of the $1 \\\\times 1$ matrix $[a]$ is $a$, and the determinant of the $2 \\\\times 2$ matrix $\\\\left[ {\\\\begin{array}{cc}\\na & b \\\\\\\\\\nc & d \\\\\\\\\\n\\\\end{array} } \\\\right] = ad - bc$; for $n \\\\ge 2$, the determinant of an $n \\\\times n$ matrix with first row or first column $a_1$ $a_2$ $a_3$ $\\\\dots$ $a_n$ is equal to $a_1C_1 - a_2C_2 + a_3C_3 - \\\\dots + (-1)^{n+1}a_nC_n$, where $C_i$ is the determinant of the $(n - 1) \\\\times (n - 1)$ matrix formed by eliminating the row and column containing $a_i$.\",\"type\":\"Precalculus\"}","display_format":"math","language":"","answer_status":"published","assets":[],"source_url":"https://github.com/hendrycks/math","history":"initial import","indexing_mode":"index","subproblems":[],"grids":[]}