AS October 2020 Paper 1 Q1
1 In this question you must show detailed reasoning.
Find \(\displaystyle\sum_{r=2}^{50}\left(\frac{1}{r - 1} - \frac{1}{r + 1}\right)\), expressing the answer as an exact fraction. [3]
| Scheme | Marks | AO |
|---|---|---|
| DR \(\displaystyle\sum_{r=2}^{50}\left(\frac{1}{r - 1} - \frac{1}{r + 1}\right)\) \(= \left(1 - \dfrac{1}{3} + \dfrac{1}{2} - \dfrac{1}{4} + \dfrac{1}{3} - \ldots + \dfrac{1}{47} - \dfrac{1}{49} + \dfrac{1}{48} - \dfrac{1}{50} + \dfrac{1}{49} - \dfrac{1}{51}\right)\) | M1 | 2.4 |
| \(= \left(1 + \dfrac{1}{2} - \dfrac{1}{50} - \dfrac{1}{51}\right)\) | A1* | 2.2a |
| \(= \dfrac{1862}{1275}\) | A1cao | 1.1 |
| [3] |
Notes
M1: enough terms to show cancellation clearly
or \(\ldots + \dfrac{1}{n - 1} - \dfrac{1}{n + 1}\)
condone \(\ldots + \dfrac{1}{r - 1} - \dfrac{1}{r + 1}\)
A1cao: dep A1*