{"status": "success", "data": {"description_md": "Let $(a_n)$ and $(b_n)$ be the sequences of real numbers such that\n\n$$(2 + i)^n = a_n + b_ni$$for all integers $n\\geq 0$, where $i = \\sqrt{-1}$. What is$$\\sum_{n=0}^\\infty\\frac{a_nb_n}{7^n}\\,?$$\n\n$\\textbf{(A) }\\frac 38\\qquad\\textbf{(B) }\\frac7{16}\\qquad\\textbf{(C) }\\frac12\\qquad\\textbf{(D) }\\frac9{16}\\qquad\\textbf{(E) }\\frac47$\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\">(a_n)</span> and <span class=\"katex--inline\">(b_n)</span> be the sequences of real numbers such that</p>&#10;<p><span class=\"katex--display\">(2 + i)^n = a_n + b_ni</span>for all integers <span class=\"katex--inline\">n\\geq 0</span>, where <span class=\"katex--inline\">i = \\sqrt{-1}</span>. What is<span class=\"katex--display\">\\sum_{n=0}^\\infty\\frac{a_nb_n}{7^n}\\,?</span></p>&#10;<p><span class=\"katex--inline\">\\textbf{(A) }\\frac 38\\qquad\\textbf{(B) }\\frac7{16}\\qquad\\textbf{(C) }\\frac12\\qquad\\textbf{(D) }\\frac9{16}\\qquad\\textbf{(E) }\\frac47</span></p>&#10;<hr/>&#10;<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>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 5, "problem_name": "2020 AMC 12A Problem 22", "can_next": true, "can_prev": true, "nxt": "/problem/20_amc12A_p23", "prev": "/problem/20_amc12A_p21"}}