Challenging +1.2 This question requires identifying square and cube numbers up to 999, applying inclusion-exclusion principle for sixth powers (counted in both), then subtracting from the arithmetic series sum. While it involves multiple steps and careful counting (31 squares, 9 cubes, 3 sixth powers), the techniques are standard for Further Maths students and the problem structure is relatively straightforward once the approach is recognized.
Question 6:
6 | Obtains or uses the sum of
the integers from 1 to 999 | 1.1b | B1 | 999
999×1000
�𝑟𝑟 = = 499500
𝑟𝑟=1 2
31
2 31×32×63
�𝑟𝑟 = = 10416
𝑟𝑟=1 6
9
2 2
3 9 ×10
�𝑟𝑟 = = 2025
𝑟𝑟=1 4
Sixth powers:
Required total:1 +64+729 = 794
499500−10416−2025+794
= 487853
Deduces that there are 31
square numbers or 9 cube
numbers between 1 and
999, inclusive | 2.2a | B1
Subtracts their sums of
squares and cubes from
the sum of integers. | 1.1a | M1
Identifies at least one of
the sixth powers (1, 64,
729) which are duplicated
in the sums of squares and
cubes | 3.1a | M1
Obtains the correct sum of
487853 | 1.1b | A1
Total | 5
Q | Marking
Instructions | AO | Marks | Typical Solution
Find the sum of all the integers from 1 to 999 inclusive that are not square or cube numbers.
[5 marks]
\hfill \mbox{\textit{AQA Further Paper 2 2020 Q6 [5]}}