| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2007 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Proof by induction |
| Type | Prove sequence property via recurrence |
| Difficulty | Moderate -0.5 This is a straightforward induction proof with an explicit formula given. Part (i) is simple algebra, and part (ii) requires only basic induction structure with the recurrence relation doing most of the work. The divisibility claim is obvious from the formula u_n = n(n+3), making this easier than a typical proof question. |
| Spec | 4.01a Mathematical induction: construct proofs |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(u_{n+1} - u_n = 2n + 4\) | B1 | Correct expression for \(u_{n+1}\) |
| M1 | Attempt to expand and simplify | |
| A1 | Obtain given answer correctly | |
| (ii) State \(u_1 = 4\) (or \(u_2 = 10\)) and is divisible by 2 | B1 | |
| State induction hypothesis true for \(u_n\) | M1 | |
| M1 | Attempt to use result in (i) | |
| A1 | Correct conclusion reached for \(u_{n+1}\) | |
| A1 | Clear, explicit statement of induction conclusion | (5 marks, 8 total) |
(i) $u_{n+1} - u_n = 2n + 4$ | B1 | Correct expression for $u_{n+1}$
| M1 | Attempt to expand and simplify
| A1 | Obtain given answer correctly
(ii) State $u_1 = 4$ (or $u_2 = 10$) and is divisible by 2 | B1 |
State induction hypothesis true for $u_n$ | M1 |
| M1 | Attempt to use result in (i)
| A1 | Correct conclusion reached for $u_{n+1}$
| A1 | Clear, explicit statement of induction conclusion | (5 marks, 8 total)6 The sequence $u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots$ is defined by $u _ { n } = n ^ { 2 } + 3 n$, for all positive integers $n$.\\
(i) Show that $u _ { n + 1 } - u _ { n } = 2 n + 4$.\\
(ii) Hence prove by induction that each term of the sequence is divisible by 2 .
\hfill \mbox{\textit{OCR FP1 2007 Q6 [8]}}