| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2008 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Proof by induction |
| Type | Suggest and prove formula |
| Difficulty | Standard +0.8 This is a standard Further Maths induction question requiring students to compute terms, spot a pattern (u_n = n²), and prove it formally. While the pattern recognition is straightforward and the induction proof follows a template structure, it's rated above average because: (1) it's from FP1 (inherently harder material), (2) it requires competent algebraic manipulation in the inductive step, and (3) students must independently formulate the conjecture before proving it. The question is well within reach of a prepared FP1 student but demands more mathematical maturity than typical C1/C2 content. |
| Spec | 4.01a Mathematical induction: construct proofs |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Obtain next terms | M1 | |
| \(u_2 = 4\), \(u_3 = 9\), \(u_4 = 16\) | A1 | 2 marks All terms correct |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(u_n = n^2\) | B1 | 1 mark Sensible conjecture made |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State conjecture is true for \(n = 1\) or \(2\) | B1 | |
| Find \(u_{n+1}\) in terms of \(n\) | M1 | |
| Obtain \((n+1)^2\) | A1 | |
| Statement of Induction conclusion | A1 | 4 marks |
# Question 8:
## Part (i)
| Answer | Mark | Guidance |
|--------|------|----------|
| Obtain next terms | M1 | |
| $u_2 = 4$, $u_3 = 9$, $u_4 = 16$ | A1 | **2 marks** All terms correct |
## Part (ii)
| Answer | Mark | Guidance |
|--------|------|----------|
| $u_n = n^2$ | B1 | **1 mark** Sensible conjecture made |
## Part (iii)
| Answer | Mark | Guidance |
|--------|------|----------|
| State conjecture is true for $n = 1$ or $2$ | B1 | |
| Find $u_{n+1}$ in terms of $n$ | M1 | |
| Obtain $(n+1)^2$ | A1 | |
| Statement of Induction conclusion | A1 | **4 marks** |
---8 The sequence $u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots$ is defined by $u _ { 1 } = 1$ and $u _ { n + 1 } = u _ { n } + 2 n + 1$.\\
(i) Show that $u _ { 4 } = 16$.\\
(ii) Hence suggest an expression for $u _ { n }$.\\
(iii) Use induction to prove that your answer to part (ii) is correct.
\hfill \mbox{\textit{OCR FP1 2008 Q8 [7]}}