Sequences — Paper 2 · IGCSE Mathematics 0580

9 8, 15, 22, 29, 36, …….. A sequence of numbers is shown above. (a) Find the 10th term of the sequence. Answer(a) [1] (b) Find the nth term of the sequence. Answer(b) [1] (c) Which term of the sequence is equal to 260? Answer(c) [1]

3 marks

Mark scheme: 9 (a) 71 1 (b) 7n + 1 1 Allow 7 x n + 1 (c) 37 1√ Their part (b) = 260 correctly solved

This question in 0580/21 Oct/Nov 2004

18 The first four terms of a sequence are For Examiner's T1 = 12 T2 = 12 + 22 T3 = 12 + 22 + 32 T4 = 12 + 22 + 32 + 42 . Use 1 (a) The nth term is given by Tn = n(n + 1)(2n + 1). 6 Work out the value of T23. Answer(a) T23 = [2] (b) A new sequence is formed as follows. U1 = T2 – T1 U2 = T3 – T2 U3 = T4 – T3 ……. (i) Find the values of U1 and U2. Answer(b)(i) U1 = and U2 = [2] (ii) Write down a formula for the nth term, Un . Answer(b)(ii) Un = [1] (c) The first four terms of another sequence are V1 = 22 V2 = 22 + 42 V3 = 22 + 42 + 62 V4 = 22 + 42 + 62 + 82. By comparing this sequence with the one in part (a), find a formula for the nth term, Vn . Answer(c) Vn = [2]

7 marks

Mark scheme: 1 18 (a) 4324 cao 2 M1 × 23 × 24 × 47 or better 6 (b) (i) 4, 9 2 B1 either correct (ii) (n + 1)2 or n2 + 2n + 1 1 2 (c) n(n+ 1)(2n + 1) oe 2 M1 recognising Vn = 4Tn 3

This question in 0580/21 Oct/Nov 2011

9 A sequence is given by u1 = 1 , u2 = 3 , u3 = 5 , u4 = 7 , … For Examiner's Use (a) Find a formula for un , the nth term. Answer(a) un = [2] (b) Find u29 . Answer(b) u29 = [1]

3 marks

Mark scheme: 9 (a) (2 n − )1 2 M1 (2 n + k ) or 2n – 1 (b) 57 or 7.55 1ft From their (a) 2 2

This question in 0580/22 Oct/Nov 2011

9 Find the nth term in each of the following sequences. 1 2 3 4 5 (a) , , , , , … 3 4 5 6 7 Answer(a) … [1] (b) 0, 3, 8, 15, 24, … Answer(b) … [2] _____________________________________________________________________________________

3 marks

Mark scheme: n 9 (a) oe final answer 1 n + 2 (b) n2–1 oe final answer 2 B1 for any quadratic in final answer 2 2 fi l

This question in 0580/21 Oct/Nov 2013

20 32 25 18 11 4 These are the fi rst 5 terms of a sequence. Find (a) the 6th term, Answer(a) … [1] (b) the n th term, Answer(b) … [2] (c) which term is equal to –332. Answer(c) … [2] __________________________________________________________________________________________

5 marks

Mark scheme: 20 (a) –3 1 (b) 39 – 7n oe 2 M1 for – 7n [+ k] (c) 53 2 M1 for their (b) = –332 shown provided their(b) is linear and their answer for (c) is a positive integer 2 2

This question in 0580/22 May/June 2014

11 (a) Here are the fi rst three terms of a sequence. U1 = 13 U2 = 13 + 23 U3 = 13 + 23 + 33 1 The n th term is given by Un = 4 n2 (n + 1)2. Work out the value of U39 . Answer(a) U39 = … [2] (b) Here are the fi rst three terms of another sequence. V1 = 23 V2 = 23 + 43 V3 = 23 + 43 + 63 By comparing this sequence with the sequence in part (a), fi nd a formula for the n th term, Vn . Answer(b) Vn = … [1] __________________________________________________________________________________________

3 marks

Mark scheme: 1 211 (a) 608 400 cao 2 M1 for × 39 × (39 + 1)2 4 (b) 2n2(n + 1)2 oe 1

This question in 0580/23 Oct/Nov 2014

20 The nth term of a sequence is an 2 + bn . (a) Write down an expression, in terms of a and b, for the 3rd term. … [1] (b) The 3rd term of this sequence is 21 and the 6th term is 96. Find the value of a and the value of b. You must show all your working. a = … b = … [4] Question 21 is printed on the next page.

5 marks

Mark scheme: 20 (a) 9a + 3b 1 (b) 36a + 6b = 96 or 9a + 3b = 21 B1 for correct method to eliminate M1 one variable a = 3 A1 If M0 A0 A0 scored SC1 for b = −2 A1 2 values satisfying 36a + 6b = 96 or 9a + 3b = 21 or if no working shown, but 2 correct answers given

This question in 0580/22 Feb/March 2016

15 7, 5, 3, 1, – 1, … (a) Find the next term in this sequence. … [1] (b) Find the nth term of the sequence. … [2]

3 marks

Mark scheme: 15 (a) – 3 1 (b) 9 – 2n oe 2 B1 for – 2n + k or dn + 9 where d ≠0 6 3 18 35 5

This question in 0580/21 May/June 2016

16 Here are the first four terms of a sequence. 23 17 11 5 (a) Find the next term. … [1] (b) Find the nth term. … [2]

3 marks

Mark scheme: 16(a) –1 1 16(b) –6n + 29 oe 2 M1 for −6n + k (any k ) or − kn + 29 ( k ≠ 0 )

This question in 0580/22 Oct/Nov 2017

3 Here is a sequence. a, 13, 9, 3, –5, –15, b, … Find the value of a and the value of b. a = … b = … [2]

2 marks

Mark scheme: 3 [a =] 15 2 B1 for each [b =] –27 or SC1 for reversed answers

This question in 0580/22 May/June 2018

22 Find an expression for the nth term of each sequence. (a) 11, 7, 3, – 1, … … [2] (b) 3, 6, 12, 24, … … [2]

4 marks

Mark scheme: 22(a) 15 − 4n final answer 2 B1 for 15 − kn or p – 4n (k ≠ 0) 22(b) 3 × 2n−1 oe final answer 2 B1 for recognition of powers of 2 such as 2k

This question in 0580/23 May/June 2018

11 These are the first five terms in a sequence. 8 11 14 17 20 (a) Find the next term. … [1] (b) Find an expression for the nth term. … [2]

3 marks

Mark scheme: 11(a) 23 1 11(b) 3n + 5 oe 2 B1 for 3n + j or kn + 5 , k ≠0

This question in 0580/21 Oct/Nov 2018

13 These are the first five terms of a sequence. - 4 2 8 14 20 Find an expression for the nth term of this sequence. … [2]

2 marks

Mark scheme: 13 6n – 10 oe 2 B1 for 6n + c or kn – 10 (k ≠ 0)

This question in 0580/22 Oct/Nov 2018

22 (a) These are the first four terms of a sequence. 5 8 11 14 (i) Write down the next term. … [1] (ii) Find an expression, in terms of n, for the nth term. … [2] (b) These are the first five terms of another sequence. 1 3 7 13 21 2 4 6 8 10 Find the next term. … [1]

4 marks

Mark scheme: 22(a)(i) 17 1 22(a)(ii) 3n + 2 oe final answer 2 B1 for 3n + k or cn + 2, c ≠ 0 22(b) 31 1 oe 12

This question in 0580/21 May/June 2019

14 Find the nth term of each sequence. 1 1 1 1 1 (a) , , , , , ... 2 4 6 8 10 … [1] (b) 1, 5, 25, 125, 625, ... … [2]

3 marks

Mark scheme: 14(a) 1 1 oe final answer 2n 14(b) 5n – 1 oe final answer 2 M1 for recognition of terms being powers of 5

This question in 0580/23 Oct/Nov 2019

5 (a) The nth term of a sequence is 60 - 8n . Find the largest number in this sequence. … [1] (b) Here are the first five terms of a different sequence. 12 19 26 33 40 Find an expression for the nth term of this sequence. … [2]

3 marks

Mark scheme: 5(a) 52 1 5(b) 7n + 5 oe final answer 2 B1 for 7n + a or bn + 5 b ≠ 0

This question in 0580/21 May/June 2020

14 (a) These are the first four terms of a sequence. 29 22 15 8 Write down the next two terms. … , … [2] (b) These are the first five terms of another sequence. 4 7 12 19 28 Find the nth term. … [2]

4 marks

Mark scheme: 14(a) 1 − 6 2 B1 for each If 0 scored, SC1 for two terms with a difference of –7 14(b) n 2 + 3 oe 2 M1 for any quadratic or second differences = 2

This question in 0580/22 Feb/March 2021

6 (a) The nth term of a sequence is n 2 + 3n . Find the first three terms of this sequence. … , … , … [2] (b) These are the first five terms of a different sequence. 25 18 11 4 - 3 Find the nth term of this sequence. … [2]

4 marks

Mark scheme: 6(a) 4 10 18 2 B1 for 2 correct 32 − 7n oe final answer 2 B1 for 32 – kn oe k ≠ 0 or j – 7n oe 6(b) or 32 − 7n seen then spoilt

This question in 0580/21 May/June 2021

16 Find the nth term of each sequence. (a) 8, 15, 34, 71, 132, … … [2] 4 5 6 (b) 2, 3, , , , … 1 4 16 64 256 … [3]

5 marks

Mark scheme: 16(a) n3 + 7 oe final answer 2 B1 for any cubic or for 3rd differences of 6 16(b) n + 1 3 B1 for n + 1 oe final answer n −1 4 B2 for 4n – 1 oe or B1 for 4n – k oe k can be 0 Maximum 2 marks if not correctly combined as a fraction

This question in 0580/21 Oct/Nov 2021

12 The table shows the first five terms of sequences A, B and C. 1st term 2nd term 3rd term 4th term 5th term nth term Sequence A 8 3 -2 -7 -12 3 4 5 6 Sequence B 2 2 3 4 5 1 Sequence C 1 2 4 8 2 Complete the table to show the nth term of each sequence. [5]

5 marks

Mark scheme: 12 13 – 5n oe final answer 5 B2 for 13 – 5n oe final answer n + 1 or B1 for – 5n + c or 13 – kn k ≠ 0 oe final answer or 13 – 5n seen then spoilt n 2n – 2 oe final answer n + 1 B1 for oe final answer n B2 for 2n –2 oe final answer or B1 for 2n – k oe k can be 0

This question in 0580/22 Oct/Nov 2021

10 These are the first four terms of a sequence. 3 -1 -5 -9 (a) Find the next term in this sequence. … [1] (b) Find the nth term. … [2]

3 marks

Mark scheme: 10(a) –13 1 10(b) –4n + 7 oe final answer 2 B1 for – 4n + k or jn + 7 (j ≠ 0) or for a correct answer spoilt

This question in 0580/23 Oct/Nov 2021

10 (a) The nth term of a sequence is n 2 + 7 . Find the first three terms of this sequence. … , … , … [2] (b) These are the first four terms of a different sequence. 15 7 − 1 -9 Find the nth term of this sequence. … [2]

4 marks

Mark scheme: 10(a) 8 11 16 2 B1 for two correct 10(b) 23 – 8n oe final answer 2 B1 for j – 8n or 23 – kn k ≠ 0 or 23 – 8n seen then spoilt

This question in 0580/22 Feb/March 2022

10 22, 17, 12, 7, 2, … (a) Find the next term of the sequence. … [1] (b) Find the nth term of the sequence. … [2]

3 marks

Mark scheme: 10(a) –3 1 10(b) 27 – 5n oe final answer 2 B1 for j – 5n or 27 – kn, k ≠0 or for 27 – 5n seen then spoilt

This question in 0580/21 May/June 2022

8 The nth term of a sequence is n 2 + 12 . Find the first three terms of this sequence. … , … , … [2]

2 marks

Mark scheme: 8 13 16 21 2 B1 for 2 correct terms in correct position or SC1 for 12, 13, 16

This question in 0580/22 May/June 2022

21 The nth term of a sequence is an 2 + bn - 4 . The first term is - 3 and the second term is 2. Find the value of a and the value of b. a = … b = … [5]

5 marks

Mark scheme: 21 [a =] 2 5 M2 for correct method to find two simultaneous [b =] − 1 equations e.g. two from a  21 b 1 4 3 a  2 2  b  2  4  2 3a + b = 2 – – 3 or M1 for 1 correct equation M1 for correctly eliminating one variable for their simultaneous equations A1 for a = 2 A1 for b = − 1

This question in 0580/22 May/June 2022

5 The nth term of a sequence is n 2 - 1. Find the first three terms of this sequence. … , … , … [2]

2 marks

Mark scheme: 5 0, 3, 8 2 B1 for 2 correct terms in correct position or SC1 for −1, 0, 3

This question in 0580/23 May/June 2022

14 Find the nth term of this sequence. 8, 17, 32, 53, 80, … … [2]

2 marks

Mark scheme: 14 3n2 + 5 oe final answer 2 M1 for correctly finding second differences or an answer that is a quadratic sequence

This question in 0580/23 May/June 2022

8 (a) 3, 9, 27, 81, … Write down the next term in this sequence. … [1] (b) 13, 17, 21, 25, … Find the nth term of this sequence. … [2]

3 marks

Mark scheme: 8(a) 243 1 8(b) 4n + 9 oe final answer 2 B1 for 4n + k or jn + 9, j ≠ 0 or for correct answer seen then spoilt

This question in 0580/21 Oct/Nov 2022

6 (a) These are the first five terms of a sequence. 27 26 23 18 11 Find the next two terms in the sequence. … , … [2] (b) The table shows information about two different sequences. First five terms of sequence nth term Sequence A 3 10 17 24 31 Sequence B 2 11 26 47 74 Complete the table. [4]

6 marks

Mark scheme: 6(a) 2 –9 2 B1 for one correct 6(b) Sequence A 2 B1 for 7n + c or kn – 4 k ≠ 0 7n – 4 oe final answer or for correct answer seen then spoilt Sequence B 2 M1 for finding second differences of 6 3n2 – 1 oe final answer or has an answer that is a quadratic sequence or for correct answer seen then spoilt

This question in 0580/22 Oct/Nov 2022

4 23, 17, 11, 5, … (a) Write down the next number in this sequence. … [1] (b) Find the nth term of this sequence. … [2]

3 marks

Mark scheme: 4(a) –1 1 4(b) 29 – 6n oe final answer 2 B1 for k – 6n or 29 – kn or 29 – 6n seen then spoiled

This question in 0580/22 Feb/March 2023

20 Find the nth term of each sequence. (a) –1, 0, 7, 26, 63, f … [2] (b) 24, 12, 6, 3, 1.5, f … [2]

4 marks

Mark scheme: 20(a) ( n − 1)3 − 1 oe 2 M1 for any cubic or third differences = 6 20(b) n−1 2 an + b  1   1  24   oe M1 for c   oe where a, b and c are  2   2  constants and a > 0

This question in 0580/22 Feb/March 2023

8 (a) The nth term of a sequence is 10 - n 2 . Write down the first three terms of this sequence. … , … , … [2] (b) These are the first four terms of another sequence. 7 10 13 16 Find an expression for the nth term of this sequence. … [2]

4 marks

Mark scheme: 8(a) 9 6 1 2 B1 for 2 correct 8(b) 3n + 4 oe final answer 2 B1 for 3n  j or kn  4 k  0 , or 3n  4 seen then spoilt

This question in 0580/21 May/June 2023

22 These are the first four terms of a sequence. 2.75 6 11.25 20 1 3 2 The nth term of this sequence is n + an + bn . 4 Calculate the value of a and the value of b. a = … b = … [5]

5 marks

Mark scheme: 22 1 5 1 3 1 2 [a =] oe B4 for n  n  3n seen 2 4 2 OR [b =] 3 1 M2 for any two of  a  b  2.75 4 1 8   4 a  2b  6 4 1 27   9 a  3b  11.25 4 1 64   16 a  4b  20 4 or M1 for one correct equation M1 for correct method to eliminate 1 variable B1 for 1 correct answer

This question in 0580/23 May/June 2023

19 Find the nth term of each sequence. (a) 11, 8, 5, 2, - 1, … … [2] (b) 1, 5, 25, 125, 625, … … [2]

4 marks

Mark scheme: 19(a) 14 – 3n oe final answer 2 B1 for 14 – kn or c – 3n or 14 – 3n seen then spoiled 19(b) 5 n −1 oe 2 B1 for 5an + b where a > 0 or 5k for any integer k > 1

This question in 0580/22 Oct/Nov 2023

8 (a) The nth term of a sequence is n 2 - 3 . Find the first three terms of this sequence. … , … , … [2] (b) These are the first five terms of a different sequence. 1 3 9 27 81 Find the nth term of this sequence. … [2]

4 marks

Mark scheme: 8(a) −2 1 6 2 B1 for any 2 correct in correct position If 0 scored SC1 for −−3 2 1 8(b) 3n−1 2 B1 for 3an + k , a ≠ 0 or 3c for any integer c>1

This question in 0580/22 Feb/March 2024

3 The table shows some information about two sequences. nth term 5th term Sequence A 60 – 4n Sequence B n2 – 300 (a) Complete the table. [2] (b) Find the smallest positive number in sequence B. … [2]

4 marks

Mark scheme: 3(a) 40 2 B1 for each –275 3(b) 24 2 B1 for 324 or 289 or 300 or 17.3…

This question in 0580/21 May/June 2024

8 These are the first five terms of a sequence. 11 18 25 32 39 Find an expression for the nth term of the sequence. … [2]

2 marks

Mark scheme: 8 7n + 4 oe final answer 2 B1 for 7n + j or kn + 4 k ≠ 0 , or 7n + 4 seen then spoilt

This question in 0580/21 May/June 2024

13 (a) A sequence has nth term 3n 2 - 1. Find the second term in this sequence. … [1] (b) The table shows the first five terms of sequences A and B. 1st term 2nd term 3rd term 4th term 5th term nth term Sequence A –6 –2 2 6 10 Sequence B 3 17 55 129 251 Complete the table to show the nth term of each sequence. [4]

5 marks

Mark scheme: 13(a) 11 1 13(b) 4n –10 oe final answer 4 B2 for 4n – 10 oe final answer or B1 for 4n + j or kn − 10 (k ≠ 0) or 4n – 10 seen then spoilt and 2n3 + 1 oe final answer B2 for 2n3 + 1 oe final answer or B1 for any cubic expression in n or 3rd difference = 12 or for correct answer seen then spoilt

This question in 0580/21 Oct/Nov 2024

1 These are the first eight terms of a sequence. c -3 -9 -15 -21 -27 -33 k Find the value of c and the value of k. c = … k = … [2]

2 marks

Mark scheme: Question Answer Marks Partial Marks 1 [c =] 3 2 B1 for each [k =] –39 or SC1 for c = –39 and k = 3

This question in 0580/22 Oct/Nov 2024

20 Find an expression for the nth term of this sequence. 1 , 1, 7, 49, 343, 2401, … 7 … [2]

2 marks

Mark scheme: 20 7n – 2 oe final answer 2 M1 for recognition of terms being powers of 7

This question in 0580/22 Oct/Nov 2024

7 These are the first 4 terms of a sequence. 11 8 5 2 (a) Find the next term of this sequence. … [1] (b) Find the nth term of this sequence. … [2]

3 marks

Mark scheme: 7(a) –1 1 7(b) 14 – 3n oe final answer 2 B1 for c – 3n or 14 – kn (k ≠ 0) or 14 – 3n seen then spoilt

This question in 0580/22 May/June 2025

16 The table shows three sequences. 1st 2nd 3rd 4th 5th nth term term term term term term Sequence 8 13 18 23 28 A Sequence 3 4 5 6 7 B 2 3 4 5 6 Sequence 1 1 1 2 4 C 4 2 Complete the table to show the nth term of each sequence. [5]

5 marks

Mark scheme: 16 5n + 3 oe final answer 2 B1 for answer 5n + j or kn + 3, k ≠ 0 or 5n + 3 seen then spoilt n + 2 1 oe final answer n + 1 2n – 3 oe final answer 2 B1 for answer 2 an + b , a  0 oe or a correct answer seen then spoilt

This question in 0580/23 Oct/Nov 2025