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  "body": "\n## Summary\n\nA linear difference equation is a recurrence in which the unknown sequence appears only linearly: $a_k(n) u_{n+k} + \\dots + a_0(n) u_n = b(n)$.\nThe coefficients may depend on $n$; when they are constant one gets a [[linear recurrence with constant coefficients|0NDVHBVBXB]].\nIts solutions have the same structure as those of a linear differential equation.\nThe solutions of the homogeneous equation form a vector space, and the general solution is one particular solution plus a homogeneous solution.\n\n## Explanations\n\n### L1 human\n\nIn a linear difference equation each term of the sequence is a weighted sum of earlier terms, plus perhaps an outside input.\nNo products or powers of the unknown terms appear.\nThe Fibonacci rule $u_{n+2} = u_{n+1} + u_n$ is linear with constant weights.\nThe rule $u_{n+1} = (n + 1) u_n$, which produces the factorials from $u_0 = 1$, is linear with a weight that changes with $n$.\nThe rule $u_{n+1} = u_n^2$ is not linear.\n",
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