A2 October 2020 Paper 1 Q1
1 Using standard summation of series formulae, determine the sum of the first \(n\) terms of the series
\((1 \times 2 \times 4) + (2 \times 3 \times 5) + (3 \times 4 \times 6) + \ldots\),
where \(n\) is a positive integer. Give your answer in fully factorised form. [6]
| Scheme | Marks | AO |
|---|---|---|
| \(\displaystyle\sum_{r=1}^{n} r(r + 1)(r + 3)\) | M1 | 3.1a |
| \(= \displaystyle\sum_{r=1}^{n} r^3 + 4\sum_{r=1}^{n} r^2 + 3\sum_{r=1}^{n} r\) | M1 | 2.5 |
| \(= \dfrac{1}{4}n^2(n + 1)^2 + \dfrac{2}{3}n(n + 1)(2n + 1) + \dfrac{3}{2}n(n + 1)\) | M1 A1 | 1.1 1.1 |
| \(= \dfrac{1}{12}n(n + 1)(3n^2 + 19n + 26)\) | M1 | 1.1 |
| \(= \dfrac{1}{12}n(n + 1)(n + 2)(3n + 13)\) | A1cao | 1.1 |
| [6] |
Notes
M1: \(r(r + 1)(r + 3)\)
M1: expanding and splitting sums
M1: substituting summation formulae
A1: correct expression
M1: drawing out \(n(n + 1)\)