A sequence \(\left\{ {{u_n}} \right\}\) satisfies the recurrence relation \({u_{n + 2}} = 2{u_{n + 1}} - 5{u_n}\) , \(n \in \mathbb{N}\) . Obtain an expression for \({u_n}\) in terms of n given that \({u_0} = 0\) and \({u_1} = 1\) .

the auxiliary equation is \({m^2} - 2m + 5 = 0\)    M1

the roots are \(1 \pm 2{\rm{i}}\)    A1

the general solution is \({u_n} = A{\left(1 + 2{\rm{i}}\right)^n} + B{\left(1 - 2{\rm{i}}\right)^n}\)     A1

substituting \({u_0} = 0\) and \({u_1} = 1\)     M1

\(A + B = 0\)

\(A(1 + 2{\rm{i}}) + B(1 - 2{\rm{i}}) = 1\)     A1

Note: Only award above A1 if both equations are correct.

solving

\(A\left(1 + 2{\rm{i}} - 1 + 2{\rm{i}}\right) = 1\)     (M1)

\(A = \frac{1}{{4{\rm{i}}}}\) or \( - \frac{{\rm{i}}}{4}\)     A1

\(B =  - \frac{1}{{4{\rm{i}}}}\) or \(\frac{{\rm{i}}}{4}\)     A1

therefore

\({u_n} = \frac{1}{{4{\rm{i}}}}{\left(1 + 2{\rm{i}}\right)^n} - \frac{1}{{4{\rm{i}}}}{\left(1 - 2{\rm{i}}\right)^n}\) or \(\frac{{\rm{i}}}{4}{\left(1 - 2{\rm{i}}\right)^n} - \frac{{\rm{i}}}{4}{\left(1 + 2{\rm{i}}\right)^n}\)     A1

[9 marks]