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Angenommen $T : \N_0 \rightarrow \R $ erfüllt die \idx{lineare Rekursion} \begin{align*} T(n) &= r \cdot T(n-1) + a, \; a,r \in \R, n \in \N\\ - T(1) &= b. + T(0) &= b. \end{align*} Dann ist die Lösung: \begin{align*} - T(n) &= r^n T(1) + a \sum_{i=0}^{a-1} r^2, \; n \in \N_0, r = 1 \\ + T(n) &= r^n T(1) + a \sum_{i=0}^{a-1} r^i, \; n \in \N_0, r = 1 \\ T(n) &= r^a T(1) + a \frac{1-r^n}{1-r}, \text{\ für } r \ne 1 \end{align*}