Question

Find $\mathit{f}\mathbf{\left(}\mathit{n}\mathbf{\right)}$when $\mathit{n}\mathbf{=}{\mathbf{4}}^{\mathbf{k}}$, where $\mathit{f}$satisfies the recurrence relation$\mathit{f}\mathbf{\left(}\mathit{n}\mathbf{\right)}\mathbf{=}\mathbf{5}\mathit{f}\mathbf{(}\mathbf{/}\mathbf{4}\mathbf{)}\mathbf{+}\mathbf{6}\mathit{n}$, with$\mathit{f}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\mathbf{=}\mathbf{1}$.

Short answer

The required expressions are:

$$\begin{gathered} f\left(n\right)=5^{{\log }_{4^{n+2}}}-6.4^{{{\log }_4}^{n+1}} \\ f\left(4^k\right)=5^{k+2}-6.4^{k+1} \end{gathered}$$

Step-by-step solution

Given expression

$$\begin{gathered} f\left(n\right)=5f(n/4)+6n \\ f\left(1\right)=1 \end{gathered}$$

Define the Recursive formula

A recursive formula is a formula that defines any term of a sequence in terms of its preceding terms.

Use the Recursive formula to find f(n)

Let and use.

Letand

(Root Characteristic equation)