A2 June 2024 Q2
2. Determine a closed form for the recurrence system\[u_1 = 4 \qquad u_2 = 6\]\[u_{n+2} = 6u_{n+1} - 9u_n \qquad (n = 1, 2, 3, \ldots)\]
| Scheme | Marks | AO |
|---|---|---|
| Aux equation is \(r^2 - 6r + 9 = 0 \Rightarrow r = \ldots\) | M1 | 1.1b |
| \((r - 3)^2 = 0 \Rightarrow r = 3\) | A1 | 1.1b |
| General form is \(u_n = (A + Bn)3^n\) | M1 | 1.1a |
| \(\left\{\begin{aligned} 4 &= 3(A + B) \\ 6 &= 9(A + 2B) \end{aligned}\right. \Rightarrow A = \ldots, B = \ldots\) | M1 | 2.1 |
| \(u_n = \left(2 - \dfrac{2}{3}n\right)3^n\) o.e \(u_n = (6 - 2n)3^{n-1}\) | A1 | 1.1b |
| (5) | ||
| (5 marks) |
Notes
M1: Forms and solves the auxiliary equation.
A1: Correct single root of 3.
M1: Selects the correct form for \(u_n\) for their roots of the equation. (If distinct real roots were found allow for \(u_n = A\alpha^n + B\beta^n\))
M1: Uses the values of \(u_1\) and \(u_2\) with the corresponding values of \(n\) used to form and uses a correct method to solve simultaneous equations to find the constants. If no method is shown, use of calculator, for solving the simultaneous equations the values must be correct for their equations.
A1: Correct closed form, isw