| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2009 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Proof by induction |
| Type | Suggest and prove formula |
| Difficulty | Standard +0.3 This is a standard FP1 induction question with a recurrence relation. Students compute terms, spot the pattern u_n = 2^n + 1, then prove by induction. The pattern recognition is straightforward (powers of 3 plus 1), and the inductive step follows mechanically from the recurrence relation. Slightly above average difficulty due to being Further Maths content, but routine within FP1. |
| Spec | 1.04e Sequences: nth term and recurrence relations4.01a Mathematical induction: construct proofs |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Attempt to find next 2 terms. Obtain correct answers. Show given result correctly | M1 A1 A1 | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| (ii) Expression involving a power of 3. Obtain correct answer | M1 A1 | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| (iii) Verify result true when \(n = 1\) or \(n = 2\) | B1 ft M1 | |
| \(u_{n+1} = 3(2(3^{n-1})+1) - 2\) | A1 | |
| \(u_{n+1} = 2(3^n) + 1\) | A1 | |
| Statement of induction conclusion | B1 | 5 |
| 10 |
**(i)** Attempt to find next 2 terms. Obtain correct answers. Show given result correctly | M1 A1 A1 | 3 |
$u_2 = 7$ $u_3 = 19$
**(ii)** Expression involving a power of 3. Obtain correct answer | M1 A1 | 2 |
$u_n = 2(3^{n-1}) + 1$
**(iii)** Verify result true when $n = 1$ or $n = 2$ | B1 ft M1 | | Expression for $u_{n+1}$ using recurrence relation
$u_{n+1} = 3(2(3^{n-1})+1) - 2$ | A1 | | Correct unsimplified answer
$u_{n+1} = 2(3^n) + 1$ | A1 | | Correct answer in correct form
Statement of induction conclusion | B1 | 5 |
| | | **10** |10 The sequence $u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots$ is defined by $u _ { 1 } = 3$ and $u _ { n + 1 } = 3 u _ { n } - 2$.\\
(i) Find $u _ { 2 }$ and $u _ { 3 }$ and verify that $\frac { 1 } { 2 } \left( u _ { 4 } - 1 \right) = 27$.\\
(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 2009 Q10 [10]}}