(Solved): a) A sequence of numbers is defined recursively as: \[ A_{n}=A_{n-1}+6 A_{n-2}, \quad A_{0}=4, \qu ...

a) A sequence of numbers is defined recursively as: \[ A_{n}=A_{n-1}+6 A_{n-2}, \quad A_{0}=4, \quad A_{1}=7 \] (i) Calculate \( A_{3} \) and \( A_{4} \). [2 marks] (ii) Solve the recurrence relation using the characteristic equation method. [5 marks] (iii) By using the Generating Function Method show that the Generating Function for the recurrence relation is given by \[ A(x)=\frac{4+3 x}{1-x-6 x^{2}} . \] [5 marks]