plumiFurther sequences and series: OCR A‑Level Further Maths questions
Every OCR A-Level Further Maths exam question on FP.9 Further sequences and series, 2018–2025, with its mark scheme and examiner’s report.
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![June 2022 AS Additional Pure Mathematics, question 3: The sequence { Un is given by U } 0 , U 2 =- and U 1 U U n 1 for n H . 1 1 = = + + - n 2 n 1 n + + (a) List the first seven terms of this sequence. [2] The Fibo](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/y53501-jun22-03-qp.webp?v=3)
![June 2022 AS Additional Pure Mathematics, question 6: The sequence { un is such that u } 7 , u 37 , u 337 , u 3337 , … . 1 = 2 = 3 = 4 = (a) Write down a first‑order recurrence system for { un . } [2] 1 n (b) By so](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/y53501-jun22-06-qp.webp?v=3)
![November 2021 Additional Pure Mathematics, question 8: (a) Solve the second-order recurrence system H 5 H 4 H with H 0 = , H 3 7 for = - 1 = n n n 2 1 + + n 0 . [5] H (b) (i) Write down the quadratic residues modulo](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/y54501-nov21-08-qp.webp?v=3)
![November 2020 AS Additional Pure Mathematics, question 5: (a) Determine the general solution of the first-order recurrence relation V 2 V n . [6] 1 = + n n + (b) Given that V 8 , find the exact value of V20 . [2] 1 =](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/y53501-nov20-05-qp.webp?v=3)
![June 2019 Additional Pure Mathematics, question 1: u " , is defined by u + n 1 1 The sequence un 2 , u 1 = and u 5 - for n 2 . 0 = = H u n n 2 - Prove that the sequence is periodic with period 5. [4]](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/y54501-jun19-01-qp.webp?v=3)
All 23 questions on Further sequences and series, as a list
- June 2025, Additional Pure Mathematics, Q8 (16 marks): (a) The members of the family of the sequences { Gn satisfy the first-order recurrence relation } G 1 25 . qG , for n 0 , 1 = - H n n + where q is a positive co
- June 2025, AS Additional Pure Mathematics, Q6 (10 marks): An investment company offers customers a three-year investment scheme. The scheme is modelled by the recurrence system 2 I I I I n and 0 000625 . 0 625 . for 0
- June 2024, Additional Pure Mathematics, Q5 (10 marks): In a conservation project in a nature reserve, scientists are modelling the population of one species of animal. The initial population of the species, P0, is 1
- June 2024, AS Additional Pure Mathematics, Q4 (5 marks): " ,, where n 4 The first five terms of the Fibonacci sequence, Fn H , are 1 F 1 , F 1 , F 2 , F = and F 3 5 = . 5 = = = 1 2 3 4 (a) Use the recurrence definitio
- June 2024, AS Additional Pure Mathematics, Q7 (12 marks): In a long-running biochemical experiment, an initial amount of 1200 mg of an enzyme is placed into a mixture. The model for the amount of enzyme present in the
- June 2023, Additional Pure Mathematics, Q7 (10 marks): ^ h for n 1 a n n 7 Binet’s formula for the nth Fibonacci number is given by Fn 0 , where a = - b H 5 2 - - and b (with 0 b) are the roots of x x 1 0 . 2 2 a =
- June 2023, AS Additional Pure Mathematics, Q2 (4 marks): " , are given by the formula 2 For all positive integers n, the terms of the sequence un 2 u 3 n 3 n 7 ( mod 10 ). = + + n (a) Show that u u for all positive in
- June 2023, AS Additional Pure Mathematics, Q6 (9 marks): When 106 of a certain type of bacteria are detected in a blood sample of an infected animal, 6 a course of treatment is started. The long-term aim of the treatm
- June 2022, Additional Pure Mathematics, Q3 (6 marks): ^ h plays a significant role in the sequence of Fibonacci 1 3 The irrational number 1 5 z = + 2 numbers given by F 0 , F 1 = and 1 F F F - for n H . 1 0 = = + n
- June 2022, Additional Pure Mathematics, Q6 (12 marks): In a national park, the number of adults of a given species is carefully monitored and controlled. The number of adults, n months after the start of this projec
- June 2022, AS Additional Pure Mathematics, Q3 (7 marks): The sequence { Un is given by U } 0 , U 2 =- and U 1 U U n 1 for n H . 1 1 = = + + - n 2 n 1 n + + (a) List the first seven terms of this sequence. [2] The Fibo
- June 2022, AS Additional Pure Mathematics, Q6 (14 marks): The sequence { un is such that u } 7 , u 37 , u 337 , u 3337 , … . 1 = 2 = 3 = 4 = (a) Write down a first‑order recurrence system for { un . } [2] 1 n (b) By so
- November 2021, Additional Pure Mathematics, Q8 (12 marks): (a) Solve the second-order recurrence system H 5 H 4 H with H 0 = , H 3 7 for = - 1 = n n n 2 1 + + n 0 . [5] H (b) (i) Write down the quadratic residues modulo
- November 2021, Additional Pure Mathematics, Q9 (10 marks): " , is given by u k 9 For each value of k the sequence of real numbers un 2 and u . 1 = 1 = 6 u n + + n For each of the following cases, either determine a valu
- November 2021, AS Additional Pure Mathematics, Q3 (6 marks): For positive integers n, the sequence of Fibonacci numbers, { Fn , starts with the terms } F 1 = , F 1 2 = , F 1 2 , … and is given by the recurrence relation F
- November 2021, AS Additional Pure Mathematics, Q8 (7 marks): A sequence { un is defined by the recurrence system } 2 a u 1 = and u 1 a for n H 1, where a is a positive constant. = - 2 u n 1 + n Determine with justificatio
- November 2020, Additional Pure Mathematics, Q8 (12 marks): u 3 " , of positive real numbers is defined by u + n 8 The sequence un 1 = and u for n H . 1 1 1 = u 2 n + + n 2 (a) Prove by induction that u 3 1 0 for all pos
- November 2020, AS Additional Pure Mathematics, Q5 (8 marks): (a) Determine the general solution of the first-order recurrence relation V 2 V n . [6] 1 = + n n + (b) Given that V 8 , find the exact value of V20 . [2] 1 =
- November 2020, AS Additional Pure Mathematics, Q7 (10 marks): In a conservation project, a batch of 100 000 tadpoles which have just hatched from eggs is introduced into an environment which has no frog population. Previou
- June 2019, Additional Pure Mathematics, Q1 (4 marks): u " , is defined by u + n 1 1 The sequence un 2 , u 1 = and u 5 - for n 2 . 0 = = H u n n 2 - Prove that the sequence is periodic with period 5. [4]
- June 2019, Additional Pure Mathematics, Q4 (10 marks): (a) Solve the second-order recurrence relation T 2 T 87 given that T 27 2 + =- 0 =- n n + and T 27 . [8] 1 = (b) Determine the value of T20. [2]
- June 2019, AS Additional Pure Mathematics, Q2 (4 marks): " , is defined by a 4 2 (a) The convergent sequence an 0 = and a 1 a for n 0 . 1 = + H n n a + n Calculate the limit of the sequence. [1] " , is defined by b k
- June 2019, AS Additional Pure Mathematics, Q4 (8 marks): " , is defined by u 2 4 The sequence un 1 = and u 1 2 u n for n 1 . = + H n 1 n + Determine un as a function of n. [8]
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