| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Session | Specimen |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Proof by induction |
| Type | Prove sequence property via recurrence |
| Difficulty | Standard +0.3 This is a structured induction proof with explicit guidance through each step. Part (i) is trivial calculation, part (ii) is straightforward algebra with the recurrence relation provided, and part (iii) applies standard induction machinery to a pre-established result. The question scaffolds the proof heavily, making it slightly easier than a typical A-level question but still requiring proper induction technique. |
| Spec | 1.04e Sequences: nth term and recurrence relations4.01a Mathematical induction: construct proofs |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(u_1 = 8\) | B1 | For correct value stated |
| Answer | Marks | Guidance |
|---|---|---|
| (ii) \(3^{2(n+1)} - 1 - (3^{2n} - 1) = 9 \times 3^{2n} - 3^{2n} = 8 \times 3^{2n}\) | B1, M1, A1 | For stating or using \(u_{n+1} = 3^{2(n+1)} - 1\) / For relevant manipulation of indices in \(u_{n+1}\) / For showing given answer correctly |
| Answer | Marks | Guidance |
|---|---|---|
| i.e. \(u_{k+1}\) is also divisible by 8, and result follows by the induction principle | B1, M1, M1, A1 | For explicit check for \(u_1\) / For induction hypothesis \(u_k\) is mult. of 8 / For obtaining and simplifying expr. for \(u_{k+1}\) / For correct conclusion, stated and justified |
**(i)** $u_1 = 8$ | B1 | For correct value stated
**Total: 1 mark**
**(ii)** $3^{2(n+1)} - 1 - (3^{2n} - 1) = 9 \times 3^{2n} - 3^{2n} = 8 \times 3^{2n}$ | B1, M1, A1 | For stating or using $u_{n+1} = 3^{2(n+1)} - 1$ / For relevant manipulation of indices in $u_{n+1}$ / For showing given answer correctly
**Total: 3 marks**
**(iii)** $u_1$ is divisible by 8, from (i)
Suppose $u_k$ is divisible by 8, i.e. $u_k = 8a$
Then $u_{k+1} = u_k + 8 \times 3^{2k} = 8(a + 3^{2k}) = 8b$
i.e. $u_{k+1}$ is also divisible by 8, and result follows by the induction principle | B1, M1, M1, A1 | For explicit check for $u_1$ / For induction hypothesis $u_k$ is mult. of 8 / For obtaining and simplifying expr. for $u_{k+1}$ / For correct conclusion, stated and justified
**Total: 4 marks**
**Question 4 Total: 8 marks**
---4 A sequence $u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots$ is defined by
$$u _ { n } = 3 ^ { 2 n } - 1$$
(i) Write down the value of $u _ { 1 }$.\\
(ii) Show that $u _ { n + 1 } - u _ { n } = 8 \times 3 ^ { 2 n }$.\\
(iii) Hence prove by induction that each term of the sequence is a multiple of 8 .
\hfill \mbox{\textit{OCR FP1 Q4 [8]}}