(a)     (i)     Write down the general solution of the recurrence relation \({u_n} + 2{u_{n - 1}} = 0,{\text{ }}n \geqslant 1\).

          (ii)     Find a particular solution of the recurrence relation \({u_n} + 2{u_{n - 1}} = 3n - 2,{\text{ }}n \geqslant 1\), in the

          form \({u_n} = An + B\), where \(A,{\text{ }}B \in \mathbb{Z}\).

          (iii)     Hence, find the solution to \({u_n} + 2{u_{n - 1}} = 3n - 2,{\text{ }}n \geqslant 1\) where \({u_1} = 7\).

(b)     Find the solution of the recurrence relation \({u_n} = 2{u_{n - 1}} - 2{u_{n - 2}},{\text{ }}n \geqslant 2\), where \({u_0} = 2,{\text{ }}{u_1} = 2\). Express your solution in the form \({2^{f(n)}}\cos \left( {g(n)\pi } \right)\), where the functions f and g map \(\mathbb{N}\) to \(\mathbb{R}\).

(a)     (i)     use of auxiliary equation or recognition of a geometric sequence     (M1)

          \({u_n} = {( - 2)^n}{u_0}\) or \( = {\text{A}}{( - 2)^n}\) or \({u_1}{( - 2)^{n - 1}}\)     A1

          (ii)     substitute suggested solution     M1

          \(An + B + 2\left( {A(n - 1) + B} \right) = 3n - 2\)     A1

          equate coefficients of powers of n and attempt to solve     (M1)

          \(A = 1,{\text{ }}B = 0\)     A1

          (so particular solution is \({u_n} = n\))

          (iii)     use of general solution = particular solution + homogeneous solution     (M1)

          \({u_n} = {\text{C}}{( - 2)^2} + n\)     A1

          attempt to find C using \({u_1} = 7\)     M1

          \({u_n} =  - 3{( - 2)^n} + n\)     A1

[10 marks]

(b)     the auxiliary equation is \({r^2} - 2r + 2 = 0\)     A1

solutions: \({r_1},{\text{ }}{r_2} = 1 \pm {\text{i}}\)     A1

general solution of the recurrence: \({u_n} = {\text{A}}{(1 + {\text{i}})^n} + {\text{B}}{(1 - {\text{i}})^n}\) or trig form     A1

attempt to impose initial conditions     M1

\(A = B = 1\) or corresponding constants for trig form     A1

\({u_n} = {2^{\left( {\frac{n}{2} + 1} \right)}} \times \cos \left( {\frac{{n\pi }}{4}} \right)\)     A1A1

[7 marks]

Total [17 marks]