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  "body": "\n## Summary\n\nThe Riccati difference equation is the first-order [[rational difference equation|T8YDEASD1H]] $x_{n+1} = (a x_n + b)/(c x_n + d)$ with $c \\ne 0$.\nIt is the iteration of a [[Möbius transformation|0SG74WBFA2]].\nA substitution turns it into a linear recurrence of order two, so it can be solved in closed form.\nIt is the discrete analogue of the [[Riccati equation|0VT2QK6X6Q]] for differential equations, which is linearized by a similar substitution.\n\n## Explanations\n\n### L1 human\n\nEach new term is a fraction with $x_n$ in the top and in the bottom.\nSuch a rule looks nonlinear, but it hides a linear rule.\nIf you write $c x_n + d$ as a ratio $y_{n+1}/y_n$, the numbers $y_n$ obey an ordinary two-step linear recurrence.\nSo the behaviour of the fraction rule is decided by the roots of a quadratic equation.\n",
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