| Exam Board | OCR |
|---|
| Module | FP1 (Further Pure Mathematics 1) |
|---|
| Year | 2007 |
|---|
| Session | June |
|---|
| Topic | Proof by induction |
|---|
3 Use the standard results for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) to show that, for all positive integers \(n\),
$$\sum _ { r = 1 } ^ { n } \left( 3 r ^ { 2 } - 3 r + 1 \right) = n ^ { 3 }$$