AS June 2025 Q4
4. A sequence \(\{u_n\}\), where \(n \geqslant 1\), satisfies the recurrence relation
\[4u_{n+1} + 2u_n = 3n - 5\]Given that \(u_1 = -2\)
(a) solve the recurrence relation, giving \(u_n\) in terms of \(n\) (6)
(b) hence determine the number of negative terms in the sequence \(\{u_n\}\). You must justify your answer. (3)
| Scheme | Marks | AO |
|---|---|---|
| Simplify to \(u_{n+1} = -\dfrac{1}{2}u_n + \dfrac{3}{4}n - \dfrac{5}{4}\) (aux equation \(m + \dfrac{1}{2} = 0 \Rightarrow\)) complementary function is \(A\left(-\dfrac{1}{2}\right)^n\) | B1 | 2.1 |
| Try \(u_n = an + b\) \(4\big(a(n+1) + b\big) + 2(an + b) = 3n - 5\) and by comparing linear and constant terms gives \(4a + 2a = 3\) \(4a + 4b + 2b = -5\) | M1 | 1.1b |
| \(a = \dfrac{1}{2}\) and \(b = -\dfrac{7}{6}\) | A1 | 1.1b |
| \(u_n = A\left(-\dfrac{1}{2}\right)^n + \dfrac{n}{2} - \dfrac{7}{6}\) | A1ft | 1.1b |
| \(u_1 = -2 \Rightarrow -2 = A\left(-\dfrac{1}{2}\right) + \dfrac{1}{2} - \dfrac{7}{6}\) | M1 | 3.4 |
| \(A = \dfrac{8}{3},\ u_n = \dfrac{8}{3}\left(-\dfrac{1}{2}\right)^n + \dfrac{n}{2} - \dfrac{7}{6}\) | A1 | 1.1b |
| (6) |
Notes
a1B1: CAO
a1M1: Correct linear form for particular solution and substituted into recurrence relation.
a1A1: CAO
a2A1ft: General solution correct or ft their values for \(a\) and \(b\) and/or C.F.
a1M1: Correct substitution.
a1A1: \(u_n =\) fully correct expression .
| Scheme | Marks | AO |
|---|---|---|
| \(u_2 = \dfrac{2}{3} + 1 - \dfrac{7}{6}\left(= \dfrac{1}{2}\right) \gt 0\) e.g Deduce all even terms must be \(\gt 0\) as first element of sum will always be positive and \(\dfrac{n}{2} \gt \dfrac{7}{6}\) for \(n \geqslant 4\) | B1 | 3.1a |
| Consider odd terms: \(u_3 = 0,\ u_5 = -\dfrac{1}{12} + \dfrac{5}{2} - \dfrac{7}{6}\left(= \dfrac{5}{4}\right) \gt 0\) Observe for all subsequent odd terms first negative element of sum is reducing and \(\left(\dfrac{n}{2} - \dfrac{7}{6}\right)\) is positive increasing, so all subsequent odd terms \(\gt 0\) | M1 | 2.1 |
| Hence there is just one negative term \(u_1\) | A1 | 2.2a |
| (3) | ||
| (9 marks) |
Notes
b1B1: Even terms \(\gt 0\)
b1M1: Odd terms after \(u_1 \geqslant 0\)
b1A1: CAO