MathDB
Recursive sequence

Source: 2019 AMC 10A #15 / AMC 12A #9

February 8, 2019
AMCAMC 10AMC 122019 AMC 12A2019 AMC 10A2019 AMCrecursion

Problem Statement

A sequence of numbers is defined recursively by a1=1a_1 = 1, a2=37a_2 = \frac{3}{7}, and an=an2an12an2an1a_n=\frac{a_{n-2} \cdot a_{n-1}}{2a_{n-2} - a_{n-1}}for all n3n \geq 3 Then a2019a_{2019} can be written as pq\frac{p}{q}, where pp and qq are relatively prime positive inegers. What is p+q?p+q ?
<spanclass=latexbold>(A)</span>2020<spanclass=latexbold>(B)</span>4039<spanclass=latexbold>(C)</span>6057<spanclass=latexbold>(D)</span>6061<spanclass=latexbold>(E)</span>8078<span class='latex-bold'>(A) </span> 2020 \qquad<span class='latex-bold'>(B) </span> 4039 \qquad<span class='latex-bold'>(C) </span> 6057 \qquad<span class='latex-bold'>(D) </span> 6061 \qquad<span class='latex-bold'>(E) </span> 8078