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  "body": "\n## Summary\n\nThe Z-transform method solves a [[linear recurrence with constant coefficients|0NDVHBVBXB]] with given initial values.\nOne applies the [[unilateral Z-transform|EPG6JPVJE7]] to the equation, solves the resulting algebraic equation for the transform of the unknown, and inverts.\nThe initial values enter by themselves through the [[shift property|JWH2KYCGBQ]].\nIt is the discrete analogue of the Laplace transform method for linear differential equations.\n\n## Explanations\n\n### L1 human\n\nA recurrence such as $y_{n+2} = y_{n+1} + y_n$ ties each term to earlier ones.\nAfter the Z-transform, each shift becomes multiplication by $z$, so the recurrence becomes an equation that one solves with ordinary algebra.\nThe answer is a rational function of $z$.\nSplitting it into simple fractions and reading each one back as a geometric sequence gives a closed formula for $y_n$.\n",
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