{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nFor every natural number $n$,\n$$0 + 1 + 2 + \\dots + n = \\sum_{k=0}^{n} k = \\frac{n(n + 1)}{2}.$$\n\n## Remarks\n\nThe number $n(n + 1)/2$ is the $n$th triangular number. For $n = 1, 2, 3, 4$ it is $1, 3, 6, 10$.\nA proof by [[induction|0ASJVEFA9J]] uses $\\frac{n(n+1)}{2} + (n + 1) = \\frac{(n+1)(n+2)}{2}$.\nAn equivalent form without division is $2 \\sum_{k=0}^{n} k = n(n + 1)$.\n\n## Explanations\n\n### L1 human\n\nWrite the sum $1 + 2 + \\dots + n$ two times, once forward and once backward.\nEach column then adds up to $n + 1$, and there are $n$ columns.\nSo two copies of the sum give $n(n + 1)$, and one copy is half of that.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "kind": "statement",
    "msc": [
      "05A19"
    ],
    "name": "Sum of the first n natural numbers",
    "role": "theorem",
    "sources": [
      {
        "locator": "Formula",
        "url": "https://en.wikipedia.org/wiki/Triangular_number"
      },
      {
        "locator": "Sum of consecutive natural numbers",
        "url": "https://en.wikipedia.org/wiki/Mathematical_induction"
      }
    ],
    "tag": "01QZNHQWYD",
    "uses": [
      "0ASJVEFA9J"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/01QZNHQWYD/",
  "links": [
    {
      "field": "uses",
      "target": "0ASJVEFA9J"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/01QZNHQWYD.md",
  "msc": [
    "05A19"
  ],
  "name": "Sum of the first n natural numbers",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/01Q/01QZNHQWYD.md",
  "status": null,
  "tag": "01QZNHQWYD",
  "uri": "m2://01QZNHQWYD",
  "url": "https://math2.tamnd.com/wiki/sum-of-the-first-n-natural-numbers/",
  "writing": {
    "tag": "01QZNHQWYD",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}