A2 June 2023 Q5
5. A sequence \(\{u_n\}\), where \(n \geqslant 0\), satisfies the second order recurrence relation
\[u_{n+2} = \frac{1}{2}\left(u_{n+1} + u_n\right) + 3 \quad \text{where} \quad u_0 = 15 \quad u_1 = 20\]| Scheme | Marks | AO |
|---|---|---|
| \(u_{n+2} = \dfrac{1}{2}\left(u_{n+1} + u_n\right) + 3\) | ||
| \(v_{n+2} + 2(n+2) = \dfrac{1}{2}\left(v_{n+1} + 2(n+1) + v_n + 2n\right) + 3\) | M1 | 2.1 |
| \(2v_{n+2} - v_{n+1} - v_n = 0\) | A1 | 1.1b |
| \(2m^2 - m - 1 = 0 \Rightarrow (2m+1)(m-1) = 0 \therefore m = -0.5\) and 1 | dM1 | 1.1b |
| \(v_n = A + B\left(-\dfrac{1}{2}\right)^n\) | A1ft | 1.1b |
| \(u_0 = 15 \Rightarrow v_0 = 15\) \(u_1 = 20 \Rightarrow v_1 = 18\) | B1 | 1.1b |
| \(A + B = 15\) \(A - \dfrac{1}{2}B = 18\) and solve for \(A\) and \(B\) | M1 | 1.1b |
| \(u_n = 17 - 2\left(-\dfrac{1}{2}\right)^n + 2n\) | A1 | 2.2a |
| (7) |
Notes
M1: Substituting \(u_n = v_n + 2n\) to obtain a second-order recurrence relation in \(v_n\) only
A1: CAO Correct homogeneous second-order recurrence relation
dM1: Solving their three-term auxiliary equation (dependent on previous M mark) (if no auxiliary equation seen, correct expression implies this mark)
A1ft: Correct general solution for \(v_n\) following their roots or complimentary function if they do not obtain homogeneous relationship.
B1: Either converting the first two given values for the sequence for \(u\) to \(v\) or for stating the corresponding general solution for \(u_n\)
M1: Setting up two equations using their solution and solving for their \(A\) and \(B\) (if correct \(A = 17\) and \(B = -2\)) (dependent on general solution of the form \(A + B(\alpha)^n\) where \(\alpha\) is \(-\dfrac{1}{2}\) or \(-2\))
A1: CAO (may be an expression without \(u_n =\) accept any equivalent form e.g. \(u_n = 17 + \left(-\dfrac{1}{2}\right)^{n-1} + 2n\))
Special Case – does not use the transformation Max 6/8
| Scheme | Marks | AO |
|---|---|---|
| \(2u_{n+2} - u_{n+1} - u_n = 6\) | M0 | |
| A0 | ||
| \(2u_{n+2} - u_{n+1} - u_n = 0\) | ||
| \(2m^2 - m - 1 = 0 \Rightarrow (2m+1)(m-1) = 0 \therefore m = -0.5\) and 1 | M1 | |
| C.F. \(u_n = A + B\left(-\dfrac{1}{2}\right)^n\) | A1ft | |
| Trial solution \(\lambda n\) \(\lambda(n+2) - \dfrac{1}{2}\lambda(n+1) - \dfrac{1}{2}\lambda n = 3\) \(\Rightarrow \lambda = 2 \quad\) P.S. \(2n\) General solution \(u_n = A + B\left(-\dfrac{1}{2}\right)^n + 2n\) | B1 | |
| \(A + B = 15\) \(A - \dfrac{1}{2}B = 18\) and solve for \(A\) and \(B\) | dM1 | |
| \(u_n = 17 - 2\left(-\dfrac{1}{2}\right)^n + 2n\) | A1 | |
| (7) | ||
| (b) As \(n \to \infty\), the terms of \(u_n\) are (approximately) given by the linear expression \({17 + 2n}\) | B1 | |
| (1) | ||
| (8 marks) |
M0: Transformation not used
A0
M1: Setting up and solving their three-term auxiliary equation (if no auxiliary equation seen, correct expression implies this mark)
A1ft: Correct complementary function for \(u_n\) following their roots
B1: Finding the correct particular solution and stating the corresponding general solution for \(u_n\)
dM1: Setting up two equations using their solution and solving for their \(A\) and \(B\) (if correct \(A = 17\) and \(B = -2\)) dependent on previous M mark (dependent on general solution of the form \(A + B(\alpha)^n + Cn\) where \(\alpha\) is \(-\dfrac{1}{2}\) or \(-2\))
A1: CAO (may be an expression without \(u_n =\) accept any equivalent form e.g. \(u_n = 17 + \left(-\dfrac{1}{2}\right)^{n-1} + 2n\))
(b) B1: Explanation that as \(n\) becomes large the terms of the sequence are (approximately) in arithmetic progression (or equivalent) dependent on correct expression for \(u_n\)
| Scheme | Marks | AO |
|---|---|---|
| As \(n \to \infty\), the terms of \(u_n\) are (approximately) given by the linear expression \({17 + 2n}\) | B1 | 2.4 |
| (1) | ||
| (8 marks) |
Notes
B1: Explanation that as \(n\) becomes large the terms of the sequence are (approximately) in arithmetic progression (or equivalent) dependent on correct expression for \(u_n\)