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  "body": "\n## Statement\n\nLet $n$ be a natural number, and let $F_n$ be the $n$-th [[Fibonacci number|0GA9K57A39]].\nLet $\\varphi = \\frac{1 + \\sqrt{5}}{2}$ be the [[golden ratio|0SC8T7548D]] and let $\\psi = \\frac{1 - \\sqrt{5}}{2}$.\nThen\n$$F_n = \\frac{\\varphi^n - \\psi^n}{\\sqrt{5}}.$$\n\n## Remarks\n\nThe proof uses the [[characteristic polynomial|0BWDJGQYST]] $x^2 - x - 1$ of the recurrence. Its roots are $\\varphi$ and $\\psi$, so $\\varphi^n$ and $\\psi^n$ are [[solutions|096SD4ERQH]], and the initial values fix the coefficients.\nBecause $|\\psi^n / \\sqrt{5}| \u003c \\frac{1}{2}$, $F_n$ is the integer nearest to $\\varphi^n / \\sqrt{5}$.\n\n## Explanations\n\n### L1 human\n\nFibonacci numbers are integers, but this formula uses $\\sqrt{5}$, and the irrational parts cancel.\nThe term with $\\psi^n$ is very small, so $F_n$ is close to $\\varphi^n / \\sqrt{5}$.\n",
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