{"status": "success", "data": {"description_md": "Let $a + ar_1 + ar_1^2 + ar_1^3 + \\cdots$ and $a + ar_2 + ar_2^2 + ar_2^3 + \\cdots$ be two different infinite geometric series of positive numbers with the same first term.  The sum of the first series is $r_1$, and the sum of the second series is $r_2$.  What is $r_1 + r_2$?\n\n$\\textbf{(A)}\\ 0\\qquad \\textbf{(B)}\\ \\frac {1}{2}\\qquad \\textbf{(C)}\\ 1\\qquad \\textbf{(D)}\\ \\frac {1 + \\sqrt {5}}{2}\\qquad \\textbf{(E)}\\ 2$\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 + ar_1 + ar_1^2 + ar_1^3 + \\cdots</span>  and  <span class=\"katex--inline\">a + ar_2 + ar_2^2 + ar_2^3 + \\cdots</span>  be two different infinite geometric series of positive numbers with the same first term.  The sum of the first series is  <span class=\"katex--inline\">r_1</span> , and the sum of the second series is  <span class=\"katex--inline\">r_2</span> .  What is  <span class=\"katex--inline\">r_1 + r_2</span> ?</p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A)}\\ 0\\qquad \\textbf{(B)}\\ \\frac {1}{2}\\qquad \\textbf{(C)}\\ 1\\qquad \\textbf{(D)}\\ \\frac {1 + \\sqrt {5}}{2}\\qquad \\textbf{(E)}\\ 2</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": "2009 AMC 12A Problem 17", "can_next": true, "can_prev": true, "nxt": "/problem/09_amc12A_p18", "prev": "/problem/09_amc12A_p16"}}