{"status": "success", "data": {"description_md": "If the value of $$\\large\\sum^{100}_{k=1}\\;\\dfrac{k\\cdot3^{k-1}}{\\prod\\limits^{k}_{i=1}(3+i)}$$ can be expressed as $1-\\dfrac{a^b\\cdot c!}{d!}$, where $\\gcd(a, b, c, d) = 1$ and $a, b, c, d \\in \\mathbb{Z}^+$, find $a+b+c+d$.", "description_html": "<p>If the value of <span class=\"katex--display\">\\large\\sum^{100}_{k=1}\\;\\dfrac{k\\cdot3^{k-1}}{\\prod\\limits^{k}_{i=1}(3+i)}</span> can be expressed as <span class=\"katex--inline\">1-\\dfrac{a^b\\cdot c!}{d!}</span>, where <span class=\"katex--inline\">\\gcd(a, b, c, d) = 1</span> and <span class=\"katex--inline\">a, b, c, d \\in \\mathbb{Z}^+</span>, find <span class=\"katex--inline\">a+b+c+d</span>.</p>&#10;", "hints_md": "It's not as scary as it looks! Expand it out and try to split each term into two fractions with different denominators.", "hints_html": "<p>It&#8217;s not as scary as it looks! Expand it out and try to split each term into two fractions with different denominators.</p>&#10;", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 3, "problem_name": "Astronomy III", "can_next": false, "can_prev": false, "nxt": "", "prev": ""}}