AS June 2019 Q2
2.
| Scheme | Marks | AO |
|---|---|---|
| \(u_{n+1} = 3u_n + 2^n \qquad n \geqslant 1\) | ||
| (aux equation \(m - 3 = 0 \Rightarrow\)) complementary function is \(A(3)^n\) | B1 | 1.1b |
| Consider a trial solution of the form \(u_n = k\left(2^n\right)\) so \(2k\left(2^n\right) = 3k\left(2^n\right) + 2^n\) | M1 | 1.1b |
| \(k = -1\) | A1 | 1.1b |
| General solution is \(u_n = A(3)^n - 2^n\) | A1 | 1.1b |
| (4) |
Notes
B1: CAO for complementary function
M1: substituting correct trial solution into the recurrence relation – allow substitution of \(u_n = k\left(2^n\right)\) into \(u_n = 3u_{n-1} + 2^{n-1}\) but not \(u_n = 3u_{n-1} + 2^n\)
A1: CAO \(k = -1\)
A1: CAO for the general solution – must include \(u_n = \ldots\)
| Scheme | Marks | AO |
|---|---|---|
| \(u_1 = u_2 \Rightarrow 3A - 2 = 9A - 4 \Rightarrow A = \ldots\) | M1 | 3.1a |
| \(u_n = 3^{n-1} - 2^n\) | A1 | 1.1b |
| (2) | ||
| (6 marks) |
Notes
M1: using the condition \(u_1 = u_2\) to calculate a value for their \(A\) \(\left(= \frac{1}{3}\right)\) - this mark is dependent on the general solution being of the form \(\pm\lambda(3)^n \pm \mu(2)^n\)
A1: CAO for the particular solution (oe) – must include \(u_n = \ldots\) - however, if neither (general nor particular) solution is given in terms of \(u_n\) then award this mark if correct expression in terms of \(n\) seen (or if both solutions are given in terms of say \(u_{n+1}\))