A2 June 2024 Paper 2 Q4
4 In this question you must show detailed reasoning.
The series \(S\) is defined as being the sum of the squares of all positive odd integers from \(1^2\) to \(779^2\).
Determine the value of \(S\). [5]
| Scheme | Marks | AO |
|---|---|---|
| \(S = \displaystyle\sum_{r=1}^{390}(2r - 1)^2\) or \(S = \displaystyle\sum_{r=0}^{389}(2r + 1)^2\) | M1 | 3.1a |
| \(\displaystyle = \sum_{r=1}^{390}\left(4r^2 - 4r + 1\right) = 4\sum_{r=1}^{390}r^2 - 4\sum_{r=1}^{390}r + \sum_{r=1}^{390}1\) or \(\displaystyle = \sum_{r=0}^{389}\left(4r^2 + 4r + 1\right) = 4\sum_{r=0}^{389}r^2 + 4\sum_{r=0}^{389}r + \sum_{r=0}^{389}1\) | M1 | 2.2a |
| \(\displaystyle\sum_{r=1}^{390}r^2 = \frac{1}{6} \times 390(390 + 1)(2 \times 390 + 1)\) or 19 849 115 or 79 396 460 | B1FT | 1.1 |
| \(\displaystyle\sum_{r=1}^{390}r = \frac{390(390 + 1)}{2}\) or 76 245 or 304 980 and \(\displaystyle\sum_{r=1}^{390}1 = 390\) | B1FT | 1.1 |
| So \(S = 4 \times 19\,849\,115 - 4 \times 76\,245 + 390\) \(= 79\,091\,870\) | A1 | 1.1 |
| [5] |
Notes
M1: DR. Correctly setting up \(S\) as a sum of odd squares (condone incorrect limit combination: allow 0 or 1 for lower and 389 or 390 or 399 or 400 for upper). Treat 399 or 400 as MR (799 for 779).
M1: Expanding and separating (with their limits). Term “1” must be dealt with correctly although condone poor notation if their 390 appropriately seen later). M0M1 is possible
B1FT: FT only 389 or 390. Correctly using the formula for the sum of squares up to and including 390 (or 389)
or 19 697 015 from 0 or 1 up to 389
B1FT: FT only 389 or 390. Correctly using the formula for the sum of integers up to and including 390 (or 389) and “sigma 1 = \(n\)” leading to their \(n\).
or 75 855 from 0 or 1 up to 389
NB B1B1 can only be attained with consistent upper limits
Alternative method
| Scheme | Marks |
|---|---|
| \(S = 1^2 + 3^2 + 5^2 + \ldots + 779^2\) \(\displaystyle = \sum_{r=1}^{779}r^2 - \left(2^2 + 4^2 + 6^2 + \ldots + 778^2\right)\) | M1 |
| \(\displaystyle\sum_{r=1}^{779}r^2 = \frac{1}{6} \times 779(779 + 1)(2 \times 779 + 1)\) \((= 157\,879\,930)\) | B1 |
| \(\displaystyle S = \sum_{r=1}^{779}r^2 - \left(2^2 + 4^2 + 6^2 + \ldots + 778^2\right)\) \(\displaystyle = \sum_{r=1}^{779}r^2 - 4\left(1^2 + 2^2 + 3^2 + \ldots + 389^2\right)\) | M1 |
| \(\displaystyle\sum_{r=1}^{389}r^2 = \frac{1}{6} \times 389(389 + 1)(2 \times 389 + 1)\) \((= 19\,697\,015)\) | B1 |
| So \(S = 157\,879\,930 - 4 \times 19\,697\,015\) \(= 79\,091\,870\) | A1 |
| [5] |
M1: Correctly setting up \(S\) and writing it as the difference between two sums of squares.
or \(\displaystyle\sum_{r=1}^{780}r^2 - \left(2^2 + 4^2 + 6^2 + \ldots + 780^2\right)\)
B1: Correctly using the formula for the sum of squares up to and including 779 (or 780)
or 158 488 330 up to 780
M1: Bringing out a factor of \(2^2\) (or 4) to convert the sum of even squares to a sum of squares. Could be separate.
B1: Correctly using the formula for the sum of squares up to and including 389 (or 390)
or 19 849 115 up to 390
NB B1B1 can only be attained for 779 and 389 or 780 and 390
A1: \(158\,488\,330 - 4 \times 19\,849\,115\)