plumiMethod of differences: AQA A‑Level Further Maths questions
AQA A-Level Further Maths past-paper questions on D.3 Method of differences, 2019–2025, with mark schemes and examiner’s reports where available. Add MadAsMaths questions to your practice, or browse them separately by topic and difficulty.
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All 12 questions on Method of differences, as a list
- June 2025, AS Paper 1, Q8 (7 marks): (a) Show that, for all positive integers r, 1 r2 – 1 (r + 1)2 = 2r + 1 r2 (r + 1)2 [1 mark] 8 (b) Hence, using the method of differences, show that n ∑ r =1 2r
- June 2024, Paper 2, Q13 (8 marks): (a) Use the method of differences to show that n ∑ r =2 1 (r – 1)r(r + 1) = 1 4 – 1 2n + 1 2(n + 1) [5 marks] 13 (b) Find the smallest integer n such that n ∑ r
- June 2024, AS Paper 1, Q9 (7 marks): (a) Show that, for all positive integers r, r + 1 r + 2 – r r + 1 = 1 (r + 1)(r + 2) [1 mark] ������������������������������������������������������������������
- June 2023, Paper 1, Q5 (6 marks): The function f is defined by f (r) = 2r(r − 2) (r 2 Z) 5 (a) Show that f (r + 1) − f (r) = r2r [2 marks] 5 (b) Use the method of differences to show that X n r=
- June 2023, AS Paper 1, Q7 (7 marks): (a) Show that, for all integers r, 1 2r – 1 – 1 2r + 1 = 2 (2r – 1)(2r + 1) [1 mark] 7 (b) Hence, using the method of differences, show that n r =1 1 (2r – 1)
- June 2022, AS Paper 1, Q9 (5 marks): (a) Show that, for r > 0, ln (r + 2) − ln r = ln 1 + 2 r [1 mark] 9 (b) Hence, using the method of differences, show that X n r=1 ln 1 + 2 r = ln 1 2 (n + a)(n
- November 2021, Paper 2, Q4 (7 marks): (a) Show that (r + 1)2 − r2 = 2r + 1 [1 mark] 4 (b) Use the method of differences to show that X n r=1 (2r + 1) = n2 + 2n [3 marks] 4 (c) Verify that using the
- November 2021, AS Paper 1, Q11 (4 marks): (a) Show that, for all positive integers r, 1 (r − 1)! − 1 r! = r − 1 r! [1 mark] 11 (b) Hence, using the method of differences, show that X n r=1 r − 1 r! = a
- November 2020, Paper 1, Q14 (6 marks): (a) Given that sinh (A + B) = sinh A cosh B + coshA sinhB express sinh (m + 1)x and sinh (m − 1)x in terms of sinhmx, coshmx, sinhx and coshx [1 mark] 14 (b) He
- November 2020, AS Paper 1, Q5 (4 marks): (a) Show that r2(r + 1)2 − (r − 1)2r2 = pr3 where p is an integer to be found. [1 mark] 5 (b) Hence use the method of differences to show that X n r=1 r3 = 1 4
- June 2019, Paper 2, Q14 (12 marks): Let Sn = X n r=1 1 (r + 1)(r + 3) where n × 1 14 (a) Use the method of differences to show that Sn = 5n2 + an 12(n + b)(n + c) where a, b and c are integers. [6
- June 2019, AS Paper 1, Q7 (5 marks): (a) Show that 1 r − 1 − 1 r + 1 × A r2 − 1 where A is a constant to be found. [1 mark] 7 (b) Hence use the method of differences to show that X n r=2 1 r2 − 1 ×
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