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  "body": "\n## Summary\n\nHigher-order differences are obtained by applying a difference operator several times.\nThe $n$-th forward difference $\\Delta_h^n f$ is the discrete analogue of the $n$-th derivative.\nThey detect polynomial behaviour, appear in Newton's interpolation formula, and express linear difference equations in operator form.\n\n## Explanations\n\n### L1 human\n\nTake a list of numbers, write down the gaps between neighbours, then the gaps between those gaps, and so on.\nEach new row is a difference of higher order.\nFor the cubes $0, 1, 8, 27, 64, 125$ the rows are $1, 7, 19, 37, 61$, then $6, 12, 18, 24$, then $6, 6, 6$.\nThe third row is constant, which reflects that $x^3$ has degree $3$.\n",
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