(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]