Let \({a_k} = f\left( {{4^k}} \right)\) and use\(n = {4^k}\).
\(\begin{array}{l}{a_k} = 5{a_{k - 1}} + 6 \cdot {4^k}{a_0}\\{a_k} = f\left( {{4^0}} \right)\\{a_k} = f(1)\\{a_k} = 1\end{array}\)
\({a_k} = 5{a_{k - 1}}\)
Let \({a_k} = r\) and \({a_{k - 1}} = 1\)
\(r = 5\)(Root Characteristic equation)
The solution of the recurrence relation is of the form \({a_n} = {\alpha _1}r_1^n + {\alpha _2}r_2^n\) when \({r_1}\)and\({r_2}\)the distinct roots of the characteristic equation.
\(a_k^{(h)} = \alpha \cdot {5^k}\)
\(F(k) = 6 \cdot {4^k}\)
If \(F(n) = \left( {{b_t}{n^t} + {b_{t - 1}}{n^{t - 1}} + \ldots + {b_1}n + {b_0}} \right){s^n}\) and \(s\) is not \({\bf{a}}\)root of the characteristic equation, then \(\left( {{p_t}{n^t} + {p_{t - 1}}{n^{t - 1}} + \ldots + {p_1}n + {p_0}} \right){s^n}\) is the particular solution.
\(a_k^{(p)} = {p_0} \cdot {4^k}\)
The particular solution needs to satisfy the recurrence relation:
\(\begin{array}{l}{a_n} = 5{a_{k - 1}} + 6 \cdot {4^k}{p_0} \cdot {4^k}\\{a_n} = 5{p_0} \cdot {4^{k - 1}} + 6 \cdot {4^k}\end{array}\)
\(\begin{array}{l} - {p_0} = 24\\{p_0} = - 24\end{array}\)
Thus, the particular solution then becomes:
\(\begin{array}{l}a_k^{(p)} = {p_0} \cdot {4^k}\\a_k^{(p)} = - 24 \cdot {4^k}\end{array}\)
The solution of the non-homogeneous recurrence relation is the sum of the solution of the homogeneous recurrence relation and the particular solution.
\(\begin{array}{l}{a_k} = a_k^{(h)} + a_k^{(p)}\\{a_k} = \alpha \cdot {5^k} - 24 \cdot {4^k}\end{array}\)
