Question

Suppose that $\mathit{f}\mathbf{\left(}\mathit{n}\mathbf{\right)}\mathbf{=}\mathit{f}\mathbf{(}\mathit{n}\mathbf{/}\mathbf{5}\mathbf{)}\mathbf{+}\mathbf{3}{\mathit{n}}^{\mathbf{2}}$when$\mathit{n}$ is a positive integer divisible by 5 , and $\mathit{f}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\mathbf{=}\mathbf{4}$. Find

Short answer

The answers are given below;

a)\(f\left( 5 \right) = 79\)

b)\(f\left( {125} \right) = 48,829\)

c) \(f\left( {3125} \right) = 30,517,579\)

Step-by-step solution

Recurrence Relation definition

A recurrence relation is an equation that recursively defines a sequence where the next term is a function of the previous terms:$\mathbf{\text{f(n) = a f(n / b) + c}}$

Apply Recurrence Relation

Compute these from the bottom up.

a) Repeatedly apply the recurrence relation

\(\begin{aligned}f(5) &= f(1) + 3 \times {5^2}\\& = 4 + 75\\ &= 79\end{aligned}\)

b) Repeatedly apply the recurrence relation

\(\begin{aligned}f(25) &= f(5) + 3 \times {25^2} + 46875\\ &= 79 + 1875 + 46875\\& = 1954\\f(125) &= f(25) + 3 \times {125^2} + 46875\\ &= 1954 + 46875\\ &= 48,829\end{aligned}\)

c) Repeatedly apply the recurrence relation

\(\begin{aligned}f(625) &= f(125) + 3 \times {625^2}\\ &= 48829 + 1171875\\& = 1220704\\f(3125) &= f(625) + 3 \times {3125^2}\\ &= 1220704 + 29296875\\ &= 30,517,579\end{aligned}\)