A2 June 2023 Paper 1 Q1
1 In this question you must show detailed reasoning.
Determine the value of \(\displaystyle\sum_{r=1}^{50} r^2(16 - r)\). [3]
| Scheme | Marks |
|---|---|
| DR \(\displaystyle\sum_1^{50} r^2(16 - r) = \sum_1^{50} 16r^2 - \sum_1^{50} r^3\) | M1 |
| \(= \dfrac{16}{6} \times 50 \times 51 \times 101 - \dfrac{1}{4}50^2 \times 51^2\) \(= (686800 - 1625625)\) | M1 |
| \(= -938825\) | A1 |
| [3] |
Notes
M1: Separating and using correct formulae for \(\sum r^3\) and \(\sum r^2\)
M1: Substituting must be seen
A1: Including all notation correct (Sigmas do not need limits)
Alternative method
| Scheme | Marks |
|---|---|
| \(\displaystyle\sum_1^{n} r^2(16 - r) = \sum_1^{50} 16r^2 - \sum_1^{50} r^3\) \(= \dfrac{16}{6}n(n + 1)(2n + 1) - \dfrac{1}{4}n^2(n + 1)^2\) \(= n(n + 1)\left(\dfrac{8}{3}(2n + 1) - \dfrac{1}{4}n(n + 1)\right)\) \(= \dfrac{n(n + 1)}{12}\left(32 + 61n - 3n^2\right)\) | M1 |
| \(= \dfrac{50 \times 51}{12}\left(32 + 61 \times 50 - 3 \times 50^2\right)\) | M1 |
| \(= -938825\) | A1 |
M1: Separating and using the correct formulae
M1: Substituting anywhere in the algebra