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  "body": "\n## Summary\n\nA recursively defined sequence is a [[sequence|0QE8STGJH2]] given by its first terms and a rule that computes each later term from the terms before it. A typical form is $a_0 = c$ and $a_{n+1} = f(n, a_n)$ for a given value $c$ and a given function $f$. The recursion theorem shows that exactly one sequence satisfies such a rule.\n\n## Remarks\n\nThe factorials satisfy $0! = 1$ and $(n+1)! = (n+1) \\cdot n!$.\nThe Fibonacci numbers satisfy $F_0 = 0$, $F_1 = 1$ and $F_{n+2} = F_{n+1} + F_n$. Here each term needs the two terms before it.\nIn analysis, the sequence $a_0 = 1$, $a_{n+1} = \\frac{1}{2}(a_n + 2/a_n)$ converges to $\\sqrt{2}$. It is [[Newton's method|07EXVNF9VP]] for $x^2 - 2$.\nThe rule that links the terms is a recurrence relation.\n\n## Explanations\n\n### L1 human\n\nA recursive rule tells you how to get the next term from the terms you already have.\nStart with $a_0 = 1$ and the rule $a_{n+1} = 2 a_n$, and you get $1, 2, 4, 8, \\dots$.\n",
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