| Exam Board | OCR |
|---|---|
| Module | Further Additional Pure (Further Additional Pure) |
| Session | Specimen |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Sequences and Series |
| Type | Second-Order Homogeneous Recurrence Relations |
| Difficulty | Standard +0.8 This is a Further Maths question on second-order recurrence relations with repeated roots (characteristic equation λ²-4λ+4=0 gives λ=2 twice), requiring knowledge of the (A+Bn)λⁿ form. Part (ii) requires proving integer values using induction or binomial expansion, which adds conceptual depth beyond routine application. Moderately challenging for Further Maths students but follows standard theory. |
| Spec | 8.01g Second-order recurrence: solve with distinct, repeated, or complex roots |
| Answer | Marks | Guidance |
|---|---|---|
| 4 | (i) | Characteristic Equation is (cid:79)2 (cid:16)4(cid:79)(cid:14)4(cid:32)0 |
| Answer | Marks |
|---|---|
| 2 n 2 | M1 |
| Answer | Marks |
|---|---|
| [4] | 1.1a |
| Answer | Marks |
|---|---|
| 1.1 | Substitution of first two known terms |
| Answer | Marks | Guidance |
|---|---|---|
| 4 | (ii) | So u (cid:32) (cid:11) 2(cid:16)n (cid:12) (cid:117)2n (cid:16) 1 |
| Answer | Marks |
|---|---|
| is an integer | M1 |
| Answer | Marks |
|---|---|
| [2] | 3.1a |
| 2.4 | e |
| m | Do not penalise lack of checking for |
Question 4:
4 | (i) | Characteristic Equation is (cid:79)2 (cid:16)4(cid:79)(cid:14)4(cid:32)0
(cid:159) (cid:79) = 2 (twice)
and General Solution is u (cid:32)(A(cid:14) Bn)(cid:117)2n
n
u (cid:32)1 and u (cid:32)1 (cid:159) 1 = A and 1 = 2(A + B)
0 1
(cid:159) A = 1, B = (cid:16) 1 and u (cid:32) (cid:11) 1(cid:16) 1n (cid:12) (cid:117)2n
2 n 2 | M1
A1
M1
A1 FT
[4] | 1.1a
1.1
1.1
1.1 | Substitution of first two known terms
n
FT correctly solved sim. eqns.
4 | (ii) | So u (cid:32) (cid:11) 2(cid:16)n (cid:12) (cid:117)2n (cid:16) 1
n
and both parts of the product are integers so u
n
is an integer | M1
E1
[2] | 3.1a
2.4 | e
m | Do not penalise lack of checking for
u = 1 since this is given
04 (i) Solve the recurrence relation $u _ { n + 2 } = 4 u _ { n + 1 } - 4 u _ { n }$ for $n \geq 0$, given that $u _ { 0 } = 1$ and $u _ { 1 } = 1$.\\
(ii) Show that each term of the sequence $\left\{ u _ { n } \right\}$ is an integer.
\hfill \mbox{\textit{OCR Further Additional Pure Q4 [6]}}