| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2009 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Proof by induction |
| Type | Prove sequence property via recurrence |
| Difficulty | Standard +0.3 This is a straightforward induction proof with a helpful part (i) that essentially provides the inductive step structure. Students need to verify the base case (trivial: u₁ = 19 = 7×1 + 6×2, divisible by 7), assume u_n is divisible by 7, then use the given identity to show u_{n+1} is also divisible by 7. The algebraic manipulation is minimal since part (i) does the heavy lifting. Slightly above average difficulty due to being Further Maths and requiring understanding of induction mechanics, but this is a textbook-standard FP1 induction question. |
| Spec | 4.01a Mathematical induction: construct proofs |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(13^n + 6^{n-1} + 13^{n+1} + 6^n\) | B1 | Correct expression seen |
| Factorisation attempt | M1 | Attempt to factorise both terms in (i) |
| Correct factorised expression | A1 | Obtain correct expression |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Check \(n=1\) (or 2) | B1 | Check that result is true for \(n=1\) (or 2) |
| Divisibility by 7 of (i) | B1 | Recognise that (i) is divisible by 7 |
| \(u_{n+1}\) divisible by 7 | B1 | Deduce that \(u_{n+1}\) is divisible by 7 |
| Induction conclusion | B1 | Clear statement of Induction conclusion |
## Question 7:
### Part (i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $13^n + 6^{n-1} + 13^{n+1} + 6^n$ | B1 | Correct expression seen |
| Factorisation attempt | M1 | Attempt to factorise both terms in (i) |
| Correct factorised expression | A1 | Obtain correct expression |
### Part (ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Check $n=1$ (or 2) | B1 | Check that result is true for $n=1$ (or 2) |
| Divisibility by 7 of (i) | B1 | Recognise that (i) is divisible by 7 |
| $u_{n+1}$ divisible by 7 | B1 | Deduce that $u_{n+1}$ is divisible by 7 |
| Induction conclusion | B1 | Clear statement of Induction conclusion |
**Total: 7 marks**
---7 It is given that $u _ { n } = 13 ^ { n } + 6 ^ { n - 1 }$, where $n$ is a positive integer.\\
(i) Show that $u _ { n } + u _ { n + 1 } = 14 \times 13 ^ { n } + 7 \times 6 ^ { n - 1 }$.\\
(ii) Prove by induction that $u _ { n }$ is a multiple of 7 .
\hfill \mbox{\textit{OCR FP1 2009 Q7 [7]}}