| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2014 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Proof by induction |
| Type | Prove sequence property via recurrence |
| Difficulty | Standard +0.8 This is a multi-step Further Maths induction question requiring students to discover a divisibility pattern from computed terms, then prove it using a non-standard recurrence relation (u_{n+1} + u_n rather than the typical u_{n+1} alone). The proof requires algebraic manipulation of the recurrence and careful inductive reasoning, making it moderately challenging but still within standard FP1 scope. |
| Spec | 4.01a Mathematical induction: construct proofs4.04e Line intersections: parallel, skew, or intersecting |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(6\), \(27\) — obtain correct values | B1 | |
| \(129\) — obtain 3rd correct value | B1 | |
| [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(3\) — state a correct value | B1ft | |
| [1] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(5^{n+1} + 2^n\) — correct expression for \(u_{n+1}\) seen | B1 | Any letter, usually \(k\) or \(n\) |
| Attempt to factorise \(u_{n+1} + u_n\) | M1 | Must deal with powers of 5 and 2 |
| Obtain correct simplified answer | A1 | |
| Clear explanation why \(u_{n+1}\) is divisible by 3 | A1 | Not \(u_{n+1} + u_n\) divisible by 3 |
| Clear statement of induction process | B1 | Provided other 4 marks earned |
| [5] |
# Question 10(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $6$, $27$ — obtain correct values | B1 | |
| $129$ — obtain 3rd correct value | B1 | |
| **[2]** | | |
---
# Question 10(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $3$ — state a correct value | B1ft | |
| **[1]** | | |
---
# Question 10(iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $5^{n+1} + 2^n$ — correct expression for $u_{n+1}$ seen | B1 | Any letter, usually $k$ or $n$ |
| Attempt to factorise $u_{n+1} + u_n$ | M1 | Must deal with powers of 5 and 2 |
| Obtain correct simplified answer | A1 | |
| Clear explanation why $u_{n+1}$ is divisible by 3 | A1 | Not $u_{n+1} + u_n$ divisible by 3 |
| Clear statement of induction process | B1 | Provided other 4 marks earned |
| **[5]** | | |
The image you've shared appears to be only the **contact/back page** of an OCR document — it contains only OCR's address, contact details, company registration information, and quality assurance logos.
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If you'd like me to extract mark scheme content, please share the **actual mark scheme pages** (typically containing tables with question numbers, expected answers, marks like M1/A1/B1, and examiner guidance).10 The sequence $u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots$ is defined by $u _ { n } = 5 ^ { n } + 2 ^ { n - 1 }$.\\
(i) Find $u _ { 1 } , u _ { 2 }$ and $u _ { 3 }$.\\
(ii) Hence suggest a positive integer, other than 1 , which divides exactly into every term of the sequence.\\
(iii) By considering $u _ { n + 1 } + u _ { n }$, prove by induction that your suggestion in part (ii) is correct.
\section*{OCR}
\hfill \mbox{\textit{OCR FP1 2014 Q10 [8]}}