A2 October 2020 Paper 1 Q2
2 Find an expression for \(1 \times 2^2 + 2 \times 3^2 + 3 \times 4^2 + \ldots + n(n + 1)^2\) in terms of \(n\). Give your answer in fully factorised form. [3]
| Scheme | Marks | AO |
|---|---|---|
| \(= \displaystyle\sum_{r=1}^{n} r(r + 1)^2 = \sum_{r=1}^{n} (r^3 + 2r^2 + r)\) \(= \displaystyle\sum_{r=1}^{n} r^3 + 2\sum_{r=1}^{n} r^2 + \sum_{r=1}^{n} r\) | M1 | 1.1a |
| \(= \dfrac{1}{4}n^2(n + 1)^2 + 2.\dfrac{1}{6}n(n + 1)(2n + 1) + \dfrac{1}{2}n(n + 1)\) | A1 | 1.1 |
| \(= \dfrac{1}{12}n(n + 1)\big(3n(n + 1) + 4(2n + 1) + 6\big)\) | ||
| \(= \dfrac{1}{12}n(n + 1)\left(3n^2 + 11n + 10\right)\) | ||
| \(= \dfrac{1}{12}n(n + 1)(n + 2)(3n + 5)\) | A1 | 1.1 |
| [3] |
Notes
M1: Correct split of terms and use of formulae
A1: Correct forms for each summation. Can be earned even if 2 is dropped
\(\dfrac{1}{12}\left(3n^4 + 14n^3 + 21n^2 + 10n\right)\) earns 2 marks
A1: Fully factorised form for this mark