{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Proof\n\n1. Let $n$ be a natural number.\n2. The node [[Fibonacci sequence|0GA9K57A39]] defines $(F_m)_{m \\ge 0}$ by $F_0 = 0$, $F_1 = 1$ and, for every natural number $m$, $F_{m+2} = F_{m+1} + F_m$.\n3. Taking $m = n$ in this defining rule gives $F_{n+2} = F_{n+1} + F_n$, which is the claim.\n\n## Idea\n\nThe Fibonacci numbers are defined by their two starting values and this rule. So the recurrence holds by definition. Its use is that, with $F_0$ and $F_1$, it fixes every later term.\n",
  "commit": "4a3290d299c8b2cd6493fce2296920ba7f82d942",
  "front": {
    "kind": "proof",
    "msc": [
      "11B39"
    ],
    "name": "Proof of the Fibonacci recurrence from the definition",
    "proves": "0D0SMQYEJA",
    "sources": [
      {
        "locator": "Definition",
        "url": "https://en.wikipedia.org/wiki/Fibonacci_sequence"
      }
    ],
    "tag": "B35K99ADCP",
    "uses": [
      "0GA9K57A39"
    ]
  },
  "kind": "proof",
  "link": "https://math2.tamnd.com/node/B35K99ADCP/",
  "links": [
    {
      "field": "proves",
      "target": "0D0SMQYEJA"
    },
    {
      "field": "uses",
      "target": "0GA9K57A39"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/B35K99ADCP.md",
  "msc": [
    "11B39"
  ],
  "name": "Proof of the Fibonacci recurrence from the definition",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/B35/B35K99ADCP.md",
  "status": null,
  "tag": "B35K99ADCP",
  "uri": "m2://B35K99ADCP",
  "url": "https://math2.tamnd.com/wiki/proof-of-the-fibonacci-recurrence-from-the-definition/",
  "writing": {
    "tag": "B35K99ADCP",
    "kind": "proof",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}