RSolve¶

Solve recurrence equations.

The general solution of a first-order linear recurrence anchors its constant at n = 1, so the root carries the exponent n - 1.

$ wo 'RSolve[a[n] == 2 a[n-1], a[n], n]'
{{a[n] -> 2^(-1 + n)*C[1]}}

With an initial condition the constant is determined.

$ wo 'RSolve[{a[n + 1] == 2 a[n], a[0] == 1}, a, n]'
{{a -> Function[{n}, 2^n]}}

An initial condition at a non-zero index shifts the exponent accordingly.

$ wo 'RSolve[{a[n] == 2 a[n-1], a[2] == 5}, a[n], n]'
{{a[n] -> 5*2^(-2 + n)}}

A repeated characteristic root contributes an extra factor of n per multiplicity.

$ wo 'RSolve[a[n] == 4 a[n-1] - 4 a[n-2], a[n], n]'
{{a[n] -> 2^n*C[1] + 2^n*n*C[2]}}