Sequences — Paper 4 · IGCSE Mathematics - International 0607

1 (a) Find the next term and the nth term in each of the following sequences. (i) 4, 8, 12, 16, 20, … next term = … nth term = … [2] (ii) -1, -3, -5, -7, -9, … next term = … nth term = … [3] (iii) 3, 12, 27, 48, 75, … next term = … nth term = … [3] (iv) 1, 8, 27, 64, 125, … next term = … nth term = … [2] (b) Use your answers to part (a), to find the next term and the nth term in the following sequence. 7, 25, 61, 121, 211, … next term = … nth term = … [3]

13 marks

Mark scheme: Question Answer Marks Partial Marks 1(a)(i) 24 2 B1 for each 4n final answer 1(a)(ii) –11 3 B1 for –11 –2n + 1 oe final answer M1 for kn + 1 (where k < 0) or – 2n + k 1(a)(iii) 108 3 B1 for 108 3n2 oe final answer M1 for kn2 [+ q] 1(a)(iv) 216 2 B1 for each n3 oe final answer 1(b) 337 3 B1 for 337 n3 + 3n2 + 2n + 1 oe final answer M1 for adding their nth terms or 3rd differences = 6 and a cubic with numerical coefficients for the answer

This question in 0607/41 May/June 2017

1 (a) These are the first four terms of a sequence. 27 20 13 6 (i) Write down the next two terms. … , … [2] (ii) Find the nth term. … [2] (b) These are the first four terms of another sequence. 8 16 32 64 (i) Write down the next two terms. … , … [2] (ii) Find the nth term. … [2]

8 marks

Mark scheme: Question Answer Marks Partial Marks 1(a)(i) – 1 2 B1 for each – 8 1(a)(ii) – 7n + 34 oe final answer 2 B1 for – 7n + k or – kn + 34 oe or correct unsimplified seen 1(b)(i) 128, 256 2 B1 for each 1(b)(ii) 2n + 2 oe final answer nfww 2 M1 for k × 2p ( k ≠ 0 ) seen, k numerical and p = f(n)

This question in 0607/42 Oct/Nov 2017

7 In this question, all lengths are measured in millimetres. 44 A small plastic cup, A, is shown in this diagram. 55 A 28 5 These plastic cups are stacked as shown in the diagram. 5 55 (a) Find the height of a stack of 8 of these cups. … mm [2] (b) Find the number of these cups in a stack that has a total height of 105 mm. … [2] (c) A similar cup, B, has base diameter 42 mm. Find the height of this cup. … mm [2] (d) 2r + 2a h 2r 2 2 r h (3r + 3ar + a ) The formula for the volume of a similar cup is V = . 3 (i) For cup A, show that a = 8 mm. [2] (ii) Find the volume of cup A. … mm3 [2] (iii) Find the volume of cup B. … mm3 [3] 2 2 r h (3r + 3ar + a ) (iv) Rearrange V = to make h the subject. 3 h = … [2]

15 marks

Mark scheme: 7(a) 90 2 M1 for 55 + 5k, k = 7 or 8 7(b) 11 2 M1 for 55 + 5(n – 1) = 105 or better 105 − 55[ + 5] or soi by 10 5 7(c) 82.5 2 42 [ ] M1 for = oe 28 55 7(d)(i) 28 + 2a = 44 oe or 44 – 28 oe seen M1 44 − 28 A1 2a = 16 or oe[= 8] 2 7(d)(ii) 56 900 or 56 900 to 56 920 2 π 2 2 M1 for × (3 × 14 + 3 × 14 × 8 + 8 ) [× 55] 3 7(d)(iii) 192 000 or 192 000 to 192 200 3 3  42  M2 for their (d)(ii) ×   28   42  3  28  3 or M1 for   or    28   42  OR M2 for π 2 2 × their (c) × 3 × 21 + 3 × (8 × 1.5) × 21 + ( 8 × 1.5 ) ( ) 3 or B1 for a =12 7(d)(iv) 3V 2 M1 for 3V = π h 3r 2 + 3ar + a 2 [ h = ] ( ) π(3r 2 + 3ar + a 2 ) V h 3V or = or πh = π 3r 2 + 3ar + a 2 3 3r 2 + 3ar + a 2 ( )

This question in 0607/42 May/June 2018

9 Pattern 1 Pattern 2 Pattern 3 Pattern 4 (a) Complete the table for the sequence of patterns above. Pattern number 1 2 3 4 5 6 Number of grey tiles 1 1 9 9 Number of white tiles 0 4 4 16 Total number of tiles 1 5 13 [3] (b) Find the number of each colour of tiles in (i) Pattern 15, Grey … White … [2] (ii) Pattern 20. Grey … White … [2] (c) Find an expression, in terms of n, for the total number of tiles in Pattern n. … [2]

9 marks

Mark scheme: 9(a) 25 25 3 B1 for each row 16 36 25 41 61 9(b)(i) 225 2 B1 for each 196 9(b)(ii) 361 2 B1 for each 400 9(c) n2 + (n – 1)2 oe 2 M1 for 2nd differences all 4 or quadratic expression

This question in 0607/43 May/June 2018

7 (a) Find an expression for the nth term for each of these sequences. (i) 80, 77, 74, 71, … … [2] (ii) 128, 64, 32, 16, … … [2] (b) The nth term of a sequence is n 2 - 1. Find the first four terms of this sequence. … , … , … , … [2] (c) The nth term of a sequence is n - 3 . Find the first four terms of this sequence. … , … , … , … [2] (d) The nth term of a sequence is n 2 + n + 41. (i) Find the first three terms of this sequence. … , … , … [2] (ii) Show that when n = 41 the number in this sequence is not prime. [1]

11 marks

Mark scheme: 7(a)(i) − 3n + 83 oe 2 B1 for − 3n + k or − kn + 83 7(a)(ii) n −1 2 k  1  8 −n  1  k − n kn + c 128   oe or 2 oe B1 for 128   or 2 or 2 oe  2   2  7(b) 0, 3, 8, 15 2 B1 for 3 correct no extras 7(c) 2, 1, 0, 1 2 B1 for 3 correct no extras 7(d)(i) 43, 47, 53 2 B1 for 2 correct no extras 7(d)(ii) 41(41 + 1 + 1) oe 1

This question in 0607/41 Oct/Nov 2018

12 Here is a sequence of patterns made using identical regular hexagons. Pattern 1 Pattern 2 Pattern 3 Pattern 4 Pattern number 1 2 3 4 5 6 Number of 1 1 13 13 white hexagons Number of grey 0 6 6 24 hexagons Total number of 1 7 19 37 61 hexagons (a) Complete the table for Pattern 5 and Pattern 6. [5] (b) The nth term of the sequence for the total number of hexagons is 3n 2 + pn + q . Find the value of p and the value of q. p = … q = … [2]

7 marks

Mark scheme: 12(a) 5 B1 for each (5) (6) 37 37 24 54 (61) 91 12(b) [p =] –3 2 B1 for each [q = ] 1

This question in 0607/41 May/June 2019

8 Find the nth term of each sequence. (a) 7, 14, 21, 28, ... … [1] (b) 10, 7, 4, 1, ... … [2] (c) 8, 16, 32, 64, ... … [2] (d) 2, 6, 12, 20, ... … [2]

7 marks

Mark scheme: 8(a) 7n oe 1 8(b) 13 – 3n oe 2 B1 for k – 3n or 13 – kn oe 8(c) 2 n + 2 oe 2 B1 for [ c×] 2n+k where c is a power of 2 and k is any integer (including 0) seen 8(d) n 2 + n oe 2 B1 for quadratic expression or for second differences = 2 seen

This question in 0607/42 May/June 2019

12 (a) y varies directly as the square root of (x + 1). y = 8 when x = 24. (i) Find the value of y when x = 15. y = … [3] (ii) Find the value of x when y = 16. x = … [2] (b) Find the next term in each of the following sequences. (i) 18, 13, 8, 3, –2, … … [1] (ii) 3, 6, 11, 18, 27, … … [1] (iii) –1000, 100, –10, 1, … … [1] (iv) 0, 0, 0, 6, 24, 60, … … [2]

10 marks

Mark scheme: 12(a)(i) 6.4 3 M2 for y =1.6 x +1 or M1 for y = k x + 1 OR 8 16 M2 for y = 25 8 y or M1 for = 25 16 12(a)(ii) 99 2 16 FT M1 for x + 1 = oe their 1.6 only FT x + 1 12(b)(i) –7 1 12(b)(ii) 38 1 12(b)(iii) –0.1 oe 1 12(b)(iv) 120 2 B1 for row of 0 6 12 18 reached or M1 for (n − 2)3 − (n − 2) or ( n − 1)( n − 2)( n − 3) oe

This question in 0607/43 May/June 2019

4 Find the n th term of each sequence. (a) 16, 25, 36, 49, 64, ... … [2] (b) 3, 10, 29, 66, 127, ... … [2] (c) 64, 32, 16, 8, 4, ... … [2]

6 marks

This question in 0607/41 May/June 2020

1 For each sequence, write down the next two terms and find an expression for the nth term. (a) 15, 11, 7, 3, - 1, ... Next two terms … , … nth term … [3] (b) 1, 2, 4, 8, 16, ... Next two terms … , … nth term … [3] (c) 4, 10, 18, 28, 40, ... Next two terms … , … nth term … [3]

9 marks

Mark scheme: Question Answer Marks Partial Marks 1(a) –5, –9 1 19 – 4n oe 2 B1 for k – 4n or 19 – kn oe 1(b) 32, 64 1 2n–1 oe 2 B1 for 2(an + b) oe a ≠ 0 1(c) 54, 70 1 n2 + 3n oe 2 B1 for an2 + bn + c a ≠ 0

This question in 0607/43 May/June 2020

6 Find the next term and the nth term in each of these sequences. (a) 125, 64, 27, 8, 1, … Next term … nth term … [3] (b) 6, 12, 20, 30, 42, … Next term … nth term … [4]

7 marks

Mark scheme: 6(a) 0 B1 (6 −n ) 3 oe B2 M1 for f ( n 3 ) 216 − 108 n + 18 n 2 − n 3 6(b) 56 B1 n 2 + 3n + 2 oe B3 M2 for n 2 + an + b , a, b numeric ≠ 0 oe or M1 for f ( n 2 ) or for common difference of 2

This question in 0607/41 Oct/Nov 2020

3 Find the next term and the nth term in each of the following sequences. (a) 13, 18, 23, 28, 33, … next term = … nth term = … [3] (b) –9, –6, –1, 6, 15, … next term = … nth term = … [3] (c) 1089, 2178, 3267, 4356, 5445, … next term = … nth term = … [2] (d) 2, –4, 8, –16, 32, … next term = … nth term = … [3]

11 marks

Mark scheme: 3(a) 38 1 5n + 8 oe 2 M1 for 5n + c or kn + 8, k ≠ 0 3(b) 26 1 2 2 M1 for any quadratic expression or 2nd n − 10 oe differences of 2 3(c) 6534 1 1089n oe 1 3(d) –64 1 n n−1 2 M1 for an expression that gives powers −−( 2 ) or 2 × ( − 2 ) oe of 2 with alternating signs. If 0 scored, SC1 for 2 ×−2n−1

This question in 0607/41 May/June 2021

5 A sequence of patterns is made using grey tiles and white tiles. Pattern 1 Pattern 2 Pattern 3 (a) Complete the table. Pattern number 1 2 3 4 n Number of grey tiles 6 10 Number of white tiles 0 2 [6] (b) Find and simplify an expression for the total number of tiles in Pattern n. … [1] (c) Pattern k has a total of 600 tiles. Find the number of grey tiles in Pattern k. … [4] (d) The tiles in a pattern are put in a bag. 5 The probability of taking a grey tile from the bag at random is . 12 A tile is taken from the bag at random and replaced. This is repeated 3 times. Find the probability that all 3 tiles are white. … [2] (e) All the grey tiles from Pattern 4 are put in a bag. Two tiles are taken from the bag at random without replacement. Find the probability that one tile came from a corner of the pattern and the other did not. … [3]

16 marks

Mark scheme: 5(a) 6 B2 for all four numbers correct 14 18 4n + 2 oe or B1 for at least two correct B2 for 4 n  2 oe 6 12 n2 – n oe or M1 for 4 n  k B2 for n 2  n oe or M1 for any quadratic or for second differences of 2 seen 5(b) n2 + 3n + 2 or (n + 1)(n + 2) 1 FT their grey + white if both in terms of n 5(c) 94 4 M1 for their k 2  3k  2 = 600 oe M1 for correct method for solving their quadratic M1 for substituting their positive integer (from a quadratic) k into their 4n+2 5(d) 343 2  5  3 oe M1 for 1  oe 1728    12  5(e) 56 3 FT their 18 from (a) for M marks only oe 153 4 14 14 4 M2 for    oe 18 17 18 17 M1 for one product

This question in 0607/41 May/June 2022

9 Find the next term and the nth term in each of the following sequences. (a) 100, 91, 82, 73, 64, ... Next term = … nth term = … [3] (b) 64, -32, 16, -8, 4, ... Next term = … nth term = … [3] (c) –1, 8, 21, 38, 59, ... Next term = … nth term = … [3]

9 marks

Mark scheme: 9(a) 55 1 109 – 9n oe 2 M1 for k – 9n 9(b) –2 1 7  n 2 M1 for c(–0.5)kor c÷(–2)k ( 2) oe 9(c) 84 1 2n2 + 3n – 6 2 M1 for any 3 term quadratic or for 2nd differences of 4

This question in 0607/42 May/June 2022

4 Complete the table for the 5th term and the nth term of each sequence. Sequence 1st term 2nd term 3rd term 4th term 5th term nth term A 3 5 7 9 B 1 8 27 64 C 1 1 1 2 4 2 D 0 2 6 12 [11]

11 marks

Mark scheme: 4 Sequence A: 11 1 2n + 1 oe final answer 2 B1 for 2n + k or for kn + 1, k ≠ 0 Sequence B: 125 1 n3 oe final answer 1 Sequence C: 4 1 2n – 3 oe final answer 2 B1 for 2n + k oe Sequence D: 20 1 n2 – n oe final answer 2 M1 for 2nd differences = 2 or for any quadratic as final answer

This question in 0607/43 Oct/Nov 2022

1 For each of these sequences, find the next term and an expression for the nth term. (a) 17 14 11 8 5 … next term … nth term … [3] 1 2 3 4 5 (b) … 2 3 4 5 6 next term … nth term … [2] (c) 4 8 16 32 64 … next term … nth term … [3] (d) - 2 5 24 61 122 … next term … nth term … [3]

11 marks

Mark scheme: Question Answer Marks Partial Marks 1(a) 2 1 20 – 3n oe final answer 2 M1 for k – 3n or for correct answer seen but then spoiled 1(b) 6 1 7 n 1 oe final answer n  1 1(c) 128 1 2n + 1 oe final answer 2 M1 for 2n + k oe 1(d) 213 1 n3 – 3 oe final answer 2 B1 for any cubic seen or M1 for third differences = 6 or for correct answer seen but then spoiled

This question in 0607/42 May/June 2023

8 Find the next term and the nth term in each of the following sequences. (a) 16, 9, 2, - 5 , -12 , … next term = … nth term = … [3] (b) 2, 8, 18, 32, 50, … next term = … nth term = … [3] (c) 1, - 3 , 5, - 7 , 9, … next term = … nth term = … [3] (d) 6, 9, 10, 9, 6, … next term = … nth term = … [3]

12 marks

Mark scheme: 8(a) –19 1 −7 n + 23 oe 2 Allow full marks for 16 + (–7)(n – 1) oe M1 for −kn + 23 or −7n + k 8(b) 72 1 2n 2 2 M1 for kn 2 or 2nd differences of 4 or –4 8(c) –11 1 ( −1) n +1 (2 n − 1) oe 2 M1 for ( −1) n +1 ( an + b ) or k (2 n − 1) oe 8(d) 1 1 − n 2 + 6n + 1 2 M1 for − an 2 + bn + c with a  0 or 2nd differences of –2 or 2

This question in 0607/41 Oct/Nov 2024

7 (a) Find the next term and the nth term for each of these sequences. (i) 19 16 11 4 next term = … nth term = … [3] (ii) 20 10 5 2.5 next term = … nth term = … [3] (b) The nth term of a sequence is 2n 2 - 3n + 1 . The kth term is 465. Work out the value of k. k = … [3]

9 marks

Mark scheme: 7(a)(i) -5 1 20 −n 2 oe 2 M1 for expression involving −n 2 or for 2nd differences of –2 or 2 seen 7(a)(ii) 1.25 oe 1 n n −1 2 n  1   1   1  40   or 20   oe M1 for an expression involving   or 2−n oe  2   2   2  7(b) 16 3 2 3  ( −3) −−4 2 ( 464) M2 oe 2  2 or ( k − 16)(2k + 29) or for a correct sketch indicating solutions or M1 for correct use of formula but with one error or for 2 k 2 − 3k − 464 = 0 or for a correct sketch

This question in 0607/43 Oct/Nov 2024