| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2006 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Sequences and series, recurrence and convergence |
| Type | Proving standard summation formulae |
| Difficulty | Moderate -0.5 This is a standard textbook derivation of the sum of squares formula using method of differences. Part (i) is telescoping series (routine), part (ii) is simple algebra expansion, and part (iii) combines these with a given formula. While it requires multiple steps and is typical Further Maths content, it's a well-known proof that follows a prescribed method with no novel insight needed—slightly easier than average overall. |
| Spec | 4.06b Method of differences: telescoping series |
| Answer | Marks | Guidance |
|---|---|---|
| Part (i) | M1, A1 | Show that terms cancel in pairs; Obtain given answer correctly |
| Part (ii) | M1, A1 | Attempt to expand and simplify; Obtain given answer correctly |
| Part (iii) | B1, B1, M1, M1, A1 | Correct \(\Sigma r\) stated; \(\Sigma 1 = n\); Consider sum of three separate terms on RHS; Required sum is LHS – two terms; Correct unsimplified expression |
| A1, 2 | Obtain given answer correctly: \(\frac{1}{2}n(n+1)(2n+1)\) |
**Part (i)** | M1, A1 | Show that terms cancel in pairs; Obtain given answer correctly
**Part (ii)** | M1, A1 | Attempt to expand and simplify; Obtain given answer correctly
**Part (iii)** | B1, B1, M1, M1, A1 | Correct $\Sigma r$ stated; $\Sigma 1 = n$; Consider sum of three separate terms on RHS; Required sum is LHS – two terms; Correct unsimplified expression
| A1, 2 | Obtain given answer correctly: $\frac{1}{2}n(n+1)(2n+1)$9 (i) Use the method of differences to show that
$$\sum _ { r = 1 } ^ { n } \left\{ ( r + 1 ) ^ { 3 } - r ^ { 3 } \right\} = ( n + 1 ) ^ { 3 } - 1$$
(ii) Show that $( r + 1 ) ^ { 3 } - r ^ { 3 } \equiv 3 r ^ { 2 } + 3 r + 1$.\\
(iii) Use the results in parts (i) and (ii) and the standard result for $\sum _ { r = 1 } ^ { n } r$ to show that
$$3 \sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 2 } n ( n + 1 ) ( 2 n + 1 )$$
\hfill \mbox{\textit{OCR FP1 2006 Q9 [10]}}