A2 June 2020 Paper 2 Q6
6 Find the sum of all the integers from 1 to 999 inclusive that are not square or cube numbers. [5 marks]
| Scheme | Marks | AO |
|---|---|---|
| Obtains or uses the sum of the integers from 1 to 999 | B1 | 1.1b |
| Deduces that there are 31 square numbers or 9 cube numbers between 1 and 999, inclusive | B1 | 2.2a |
| Subtracts their sums of squares and cubes from the sum of integers. | M1 | 1.1a |
| Identifies at least one of the sixth powers (1, 64, 729) which are duplicated in the sums of squares and cubes | M1 | 3.1a |
| Obtains the correct sum of 487853 | A1 | 1.1b |
| (5 marks) |
Typical solution
\[\sum_{r=1}^{999} r = \frac{999 \times 1000}{2} = 499500\]\[\sum_{r=1}^{31} r^2 = \frac{31 \times 32 \times 63}{6} = 10416\]\[\sum_{r=1}^{9} r^3 = \frac{9^2 \times 10^2}{4} = 2025\]Sixth powers: \(1 + 64 + 729 = 794\)
Required total:
\[499500 - 10416 - 2025 + 794 = 487853\]