Sequences — Paper 4 · IGCSE Mathematics 0580

11 On Monday, Ankuri sent this text message to two friends. Today is Day Number 1. Tomorrow, please add 1 to the Day Number and send this text message to two friends. All the friends who receive a text message follow the instructions. (a) Complete the table. Day Monday Tuesday Wednesday Thursday Friday Saturday Sunday Day Number 1 2 3 Number of text messages sent 2 4 today [4] (b) Write down an expression for the number of text messages sent on Day Number n. … [1] (c) Ankuri thinks that, by the end of Day Number 3, the total number of text messages that have been sent is 2 4 - 2 . (i) Show that she is correct. [2] (ii) Complete the statement. The total number of text messages sent by the end of Day Number 5 is … which is equal to 2k – 2 where k = … . [2] (iii) Write down an expression for the total number of text messages sent by the end of Day Number n. … [1] (iv) Find the Day Number when the total number of text messages sent by the end of the day is 1022. … [1]

11 marks

This question in 0580/42 Feb/March 2017

9 (a) The nth term of a sequence is 8n - 3 . (i) Write down the first two terms of this sequence. … , … [1] (ii) Show that the number 203 is not in this sequence. [2] (b) Find the nth term of these sequences. (i) 13, 19, 25, 31, … … [2] (ii) 4, 8, 14, 22, … … [2] (c) … , 20, 50, … The second term of this sequence is 20 and the third term is 50. The rule for finding the next term in this sequence is subtract y then multiply by 5. Find the value of y and work out the first term of this sequence. y = … First term = … [4]

11 marks

Mark scheme: 9(a)(i) 5 and 13 1 9(a)(ii) 8n – 3 = 203 M1 Evaluation of 25th or 26th term with supporting evidence or explanation 3 A1 Second evaluation of 25th or 26th terms 25.75 or 25 with supporting evidence or explanation 4 If zero scored, SC1 for 25.75 or 197 and 205 with partial evidence or explanation 9(b)(i) 6n + 7 oe final answer 2 B1 for 6n + c or kn + 7 k ≠ 0 9(b)(ii) n2 + n + 2 oe final answer 2 B1 for a quadratic expression or second difference = 2 9(c) [y = ] 10 2 M1 for 5(20 – y) = 50 [First term = ] 14 2 M1 for 5(x – their y) = 20 or for 20 ÷ 5 + their y

This question in 0580/41 May/June 2017

11 The table shows the first four terms in sequences A, B, C and D. Complete the table. Sequence 1st term 2nd term 3rd term 4th term 5th term nth term A 16 25 36 49 B 5 8 11 14 C 11 17 25 35 3 4 5 6 D 2 3 4 5 [12]

12 marks

Mark scheme: 11 2 1, 2 M1 for a quadratic expression seen or 64 ( n + 3 ) oe final answer second differences 2 17 3n + 2 oe final answer 1, 2 B1 for 3n + k (any k) or kn + 2 (k ≠ 0) 2 1, 2FT FT their difference expressions A – B 47 ( n + 3 ) − (3n + 2) oe isw M1 for expression an2 + bn + c seen or second differences 2 7 n + 2 1, 2 n + k + 1 oe final answer B1 for seen 6 n + 1 n + k

This question in 0580/43 May/June 2017

6 Diagram 1 Diagram 2 Diagram 3 Diagram 4 These are the first four diagrams in a sequence. Each diagram is made from small squares and crosses. (a) Complete the table. Diagram 1 2 3 4 5 n Number of crosses 6 10 14 Number of small squares 2 5 10 [6] (b) Find the number of crosses in Diagram 60. … [1] (c) Which diagram has 226 squares? Diagram … [1] (d) The side of each small square has length 1 cm. The number of lines of length 1 cm in Diagram n is 2n2 + 2n + q. Find the value of q. q = … [2]

10 marks

Mark scheme: 6(a) 18 22 4n + 2 oe 6 B2 for 18, 22, 17, 26 17 26 n2 + 1 oe or B1 for two or three correct values AND B2 for 4n + 2 oe or B1 for 4n + k oe or pn + 2 (p ≠ 0) AND B2 for n2 + 1 oe or B1 for n2 + k oe 6(b) 242 1 FT their 4n + 2 provided a linear expression 6(c) 15 1 6(d) 3 2 M1 for 2 × 12 + 2 × 1 + q = 7 oe

This question in 0580/41 Oct/Nov 2017

12 Marco is making patterns with grey and white circular mats. Pattern 1 Pattern 2 Pattern 3 Pattern 4 The patterns form a sequence. Marco makes a table to show some information about the patterns. Pattern number 1 2 3 4 5 Number of grey mats 6 9 12 15 Total number of mats 6 10 15 21 (a) Complete the table for Pattern 5. [2] (b) Find an expression, in terms of n, for the number of grey mats in Pattern n. … [2] (c) Marco makes a pattern with 24 grey mats. Find the total number of mats in this pattern. … [2] (d) Marco needs a total of 6 mats to make the first pattern. He needs a total of 16 mats to make the first two patterns. 1 3 2 He needs a total of n + an + bn mats to make the first n patterns. 6 Find the value of a and the value of b. a = … b = … [6]

12 marks

Mark scheme: 12(a) 18 2 B1 for each 28 12(b) 3n + 3 oe 2 B1 for 3n + k oe or cn + 3 oe c ≠ 0 12(c) 45 2 M1 for identifying 7th pattern or M1 for their ( 3n + 3 ) = 24 12(d) 3 13 6 M1 for any correct substitution [ a = ] oe [b = ] oe 1 2 3 e.g. (2)³ + 2²a + 2b 6 A1 for one of e.g. 1 + a + b = 6 oe 6 8 + 4a + 2b = 16 oe 6 27 + 9a + 3b = 31 oe 6 64 + 16 a + 4b = 52 oe 6 A1 for another of the above M1 for correctly eliminating one variable from their equations 3 A1 for a = 2 13 A1 for b = oe 3

This question in 0580/41 May/June 2018

10 (a) Find the next term and the nth term of this sequence. 3 4 5 6 7 , , , , , … 5 7 9 11 13 Next term = … nth term = … [3] (b) Find the nth term of each sequence. (i) –1, –3, –5, –7, –9, … … [2] (ii) 2, 9, 28, 65, 126, … … [2]

7 marks

Mark scheme: 10(a) 8 B1 15 n + 2 B2 B1 for n + 2 as numerator or 2n + 3 as oe denominator 2 n + 3 10(b)(i) 1 − 2 n oe 2 B1 for −2 n + k oe or pn + 1 (p ≠ 0) oe 10(b)(ii) n 3 + 1 oe 2 M1 for cubic expression

This question in 0580/43 Oct/Nov 2018

11 (a) The table shows the first five terms of sequence A and sequence B. Term 1 2 3 4 5 6 Sequence A 7 13 23 37 55 Sequence B 1 3 9 27 81 (i) Complete the table for the 6th term of each sequence. [2] (ii) Find the nth term of (a) sequence A, … [2] (b) sequence B. … [2] (b) The nth term of another sequence is 4n 2 + n + 3 . Find (i) the 2nd term, … [1] (ii) the value of n when the nth term is 498. n = … [3]

10 marks

Mark scheme: 11(a)(i) 77 243 2 B1 for each 11(a)(ii)(a) 2 n 2 + 5 oe 2 M1 for a quadratic expression as the answer or B1 for common 2nd difference of 4 11(a)(ii)(b) 3n −1 oe 2 B1 for 3k oe where k is a linear function of n 11(b)(i) 21 1 11(b)(ii) 11 3 B2 for ( 4 n + 45 )( n − 11) seen or B1 for 4n2 + n + 3 = 498 oe

This question in 0580/42 Feb/March 2019

10 (a) Complete the table for the 5th term and the nth term of each sequence. 1st 2nd 3rd 4th 5th nth term term term term term term 9 5 1 -3 4 9 16 25 1 8 27 64 8 16 32 64 [11] (b) 0, 1, 1, 2, 3, 5, 8, 13, 21, … This sequence is a Fibonacci sequence. After the first two terms, the rule to find the next term is “add the two previous terms”. For example, 5 + 8 = 13 . Use this rule to complete each of the following Fibonacci sequences. 2 4 … … … 1 … … … 11 … -1 … … 1 [3] 1 3 4 7 11 (c) , , , , , … 3 4 7 11 18 p (i) One term of this sequence is . q Find, in terms of p and q, the next term in this sequence. … [1] (ii) Find the 6th term of this sequence. … [1]

16 marks

Mark scheme: 10(a) – 7 11 B1 13 – 4n oe B2 or B1 for 13 – kn (k ≠ 0) or for k – 4n 36 B1 (n + 1)2 oe B2 or B1 for any quadratic 125 B1 n 3 oe B1 B1 128 2n + 2 oe B2 or B1 for 2k oe 10(b) … , … , 6, 10, 16 3 B1 for each correct row … , 3, 4, 7, … 2, … , 1, 0, … 10(c)(i) q 1 p + q 10(c)(ii) 18 1 29

This question in 0580/41 Oct/Nov 2019

7 (a) Naga has n marbles. Panav has three times as many marbles as Naga. Naga loses 5 marbles and Panav buys 10 marbles. Together they now have more than 105 marbles. Write down and solve an inequality in n. … [3] (b) y is inversely proportional to x2. When x = 4 , y = 7.5 . Find y when x = 5 . y = … [3] (c) Find the nth term of each sequence. (i) 4 2 0 - 2 - 4 … … [2] (ii) 1 7 17 31 49 … … [2]

10 marks

Mark scheme: 7(a) n − 5 + 3n + 10 > 105 or better B1 n > 25 final answer B2 M1 for 4n > 100 7(b) 4.8 3 k M1 for y = or better x 2 their k M1 for [ y = ] 5 2 OR M2 for y × 5 2 = 7.5 × 4 2 7(c)(i) 6 − 2n oe final answer 2 B1 for answer 6 − kn (k ≠ 0) oe or answer j−2n oe or for correct expression shown in working and then spoilt 7(c)(ii) 2 n 2 − 1 oe final answer 2 B1 for 2nd diff = 4 or a quadratic expression or for correct expression shown in working and then spoilt

This question in 0580/42 Feb/March 2020

7 Diagram 1 Diagram 2 Diagram 3 Diagram 4 These are the first four diagrams of a sequence. The diagrams are made from white dots and black dots. (a) Complete the table for Diagram 5 and Diagram 6. Diagram 1 2 3 4 5 6 Number of white dots 1 4 9 16 Number of black dots 0 1 3 6 Total number of dots 1 5 12 22 [2] (b) Write an expression, in terms of n, for the number of white dots in Diagram n. … [1] 1 2 (c) The expression for the total number of dots in Diagram n is ( 3n - n ) . 2 (i) Find the total number of dots in Diagram 8. … [1] (ii) Find an expression for the number of black dots in Diagram n. Give your answer in its simplest form. … [2] (d) T is the total number of dots used to make all of the first n diagrams. T = an 3 + bn 2 Find the value of a and the value of b. You must show all your working. a = … b = … [5]

11 marks

Mark scheme: 7(a) 25 36 2 B1 for 3, 4 or 5 correct 10 15 35 51 7(b) n2 1 7(c)(i) 92 1 7(c)(ii) 1 2 1 (n2 – n) oe M1 for (3n2 – n) – n2 oe 2 2 1 or for final quadratic answer with n2 oe 2 1 2 or − n oe but not both 2 7(d) 1 1 5 B2 for 2 correct equations eg a = , b = a + b = 1, 8a + 4b = 6 2 2 or B1 for 1 correct equation B2 for one correct value or M1 (dep on at least B1) for correctly eliminating one variable from two linear equations in a and b OR 1 B2 for a = 2 or B1 for 6a = 3 or for 3rd difference = 3 1 B2 for b = 2 or M1 for substituting their a into a correct equation of first differences

This question in 0580/41 Oct/Nov 2020

11 Sequence 1st term 2nd term 3rd term 4th term 5th term nth term A 13 9 5 1 B 0 7 26 63 7 8 9 10 C 8 16 32 64 (a) Complete the table for the three sequences. [10] (b) One term in Sequence C is p. q Write down the next term in Sequence C in terms of p and q. … [2]

12 marks

Mark scheme: 11(a) A : –3 17 – 4n oe 3 B1 for –3 B2 for 17 – 4n oe or B1 for k – 4n oe or 17 – pn oe, p ≠ 0 B : 124 n3 – 1 oe 3 B1 for 124 B2 for n3 – 1 oe or B1 for any cubic 11 n + 6 4 11 C : oe B1 for 128 2 n + 2 128 n + 6 B3 for oe 2 n + 2 or B2 for 2 n + 2 oe seen or B1 for 2k oe or n + 6 seen 11(b) p + 1 2 B1 for p + 1 or 2q oe oe 2 q

This question in 0580/42 Oct/Nov 2020

11 The table shows the first four terms in sequences A, B, and C. Sequence 1st term 2nd term 3rd term 4th term 5th term nth term A 4 9 14 19 B 3 10 29 66 C 1 4 16 64 Complete the table. [9]

9 marks

This question in 0580/43 Oct/Nov 2020

10 The table shows four sequences A, B, C and D. Sequence 1st term 2nd term 3rd term 4th term 5th term nth term A 1 8 27 64 B 5 11 17 23 C 0.25 0.5 1 2 4 D 4.75 10.5 16 21 Complete the table. [9]

9 marks

Mark scheme: 10 125 n3 oe final ans B2 B1 for 125 B1 for n3 29 6n – 1 oe final ans B3 B1 for 29 B2 for 6n – 1 oe or B1 for 6n + k or an – 1 (a ≠ 0) 2n – 3 oe final ans B2 B1 for 2n [+ k] oe 25 6n – 1 – 2n – 3 oe final ans B2 FT their 29 – 4 and their 6n – 1 – their 2n – 3 OR B1FT for each 1 3 1 2 17 OR 25.25 − n + n + n − 1 oe B1 for each 24 8 3 final ans

This question in 0580/42 May/June 2021

11 (a) These are the first four terms of a sequence. 11 7 3 -1 (i) Write down the next term. … [1] (ii) Write down the term to term rule for this sequence. … [1] (iii) Find the nth term of this sequence. … [2] 2n (b) The nth term of a different sequence is . n + 1 (i) Find the difference between the 5th term and the 6th term of this sequence. Give your answer as a fraction. … [2] 3 (ii) Is a term in this sequence? 4 Show how you decide. [3]

9 marks

Mark scheme: 11(a)(i) –5 1 11(a)(ii) Subtract 4 oe 1 11(a)(iii) 15 – 4n oe final answer 2 B1 for k – 4n or 15 – jn j ≠ 0 11(b)(i) 1 2 12 10 or equivalent fraction B1 for and 21 7 6 11(b)(ii) 3 M2 3 2 n n = oe M1 for = oe 5 4 n + 1 or or 2n ⩾ n + 1 but 3 < 4. M1 for 2n > n + 1 but 3 < 4 No, n is not an integer oe A1 or 3 No, is less than 1, oe 4

This question in 0580/43 May/June 2021

2 (a) Simplify fully. (i) p 3 # p 11 … [1] 18 m 6 (ii) 2 3m … [2] 1 27x 9 y 27 - 3 (iii) e 64 o … [3] (b) A sequence has nth term 3n 2. Write down the first 3 terms of this sequence. … , … , … [2] (c) Find the nth term for each of these sequences. (i) 13, 16, 19, 22, 25, … … [2] (ii) 3, 17, 55, 129, 251, … … [2] (d) Solve. 3x - 22 = 23 4 x = … [3] (e) Use the quadratic formula to solve 3x 2 + 8x - 20 = 0 . Show all your working and give your answers correct to 2 decimal places. x = … , x = … [4]

19 marks

Mark scheme: 2(a)(i) p14 final answer 1 2(a)(ii) 6m4 final answer 2 B1 for 6mk or km4 in final answer or correct answer seen and spoilt 2(a)(iii) 4 4 x −3 y −9 3 B2 for correct answer seen and spoilt or final answer or 2 correct elements in final answer 3 9 3x y 3 4 3 or B1 for one of or oe or x3 or y9 seen 3 4 2(b) 3, 12, 27 2 B1 for 12 or 27 2(c)(i) 3n + 10 oe final answer 2 B1 for 3n + k oe or jn + 10 oe (j ≠ 0) or for correct expression shown in working and then spoilt 2(c)(ii) 2n3 + 1 oe final answer 2 B1 for 3rd diff = 12 (both needed) or for cubic answer or for correct expression shown in working and then spoilt 2(d) 38 3 M2 for 3x = 4 × 23 + 22 or M1 for 3x – 22 = 4 × 23 3 x 22 or for = 23 + oe 4 4 2(e) 2 B2 2 −8 8 − 4(3)( −20) B1 for 8 − 4(3)( −20) oe 2  3 −+8 q −−8 q 2 or oe or oe or both −8 8 ( −20) 2  3 2  3 or  − 2  3 4  32 3 or better – 4.24, 1.57 final answers B2 B1 for each If B0, SC1 for answers – 4.2 or –4.23 or –4.240 to – 4.239 and 1.6 or 1.572 to 1.573 or – 4.24 and 1.57 seen in working or for –1.57 and 4.24 as final answer

This question in 0580/43 Oct/Nov 2022

6 (a) A sequence has nth term . 2n + 3 (i) Find the first three terms of this sequence. Give your answers as fractions. … , … , … [2] 12 (ii) The kth term of this sequence is . 25 Find the value of k. k = … [2] (b) Find the nth term of each sequence. (i) 6, 13, 32, 69, 130, … … [2] (ii) 100, 50, 25, 12.5, 6.25, … … [2]

8 marks

Mark scheme: 6(a)(i) 1 2 3 2 B1 for 2 correct terms isw , , final answer or for 0.2 and (0.286 or 0.2857…) and 5 7 9 0.333… 6(a)(ii) 36 2 12(2 k  3) M1 for k = or better 25 6(b)(i) n3 + 5 oe final answer 2 B1 for any cubic or common third differences of 6 (at least 2) or for correct answer seen and spoilt 6(b)(ii) 100 × 21–n oe final answer 2 n   k  [+k]  1  B1 for 2–n oe or   oe in answer  2  or for correct answer seen and spoilt

This question in 0580/42 May/June 2023

6 (a) Sequence 1st term 2nd term 3rd term 4th term 5th term nth term A -7 -3 1 5 B 7 13 23 37 C 2 3 4 5 27 81 243 729 Complete the table for the three sequences. [10] (b) In a sequence, the sum of the first 49 terms is 7644. The sum of the first 50 terms is 7975. Find the 50th term of this sequence. … [1]

11 marks

Mark scheme: 6(a) A 9 B1 4n – 11 oe final answer B2 B1 for 4n – k or jn – 11 oe j  0 B 55 B1 2n2 + 5 oe final answer B2 B1 for any quadratic or second differences = 4 6 B1 C oe 2187 n + 1 B3 B2 for 3n + 2 oe oe final answer 3n + 2 OR B1 for 3n + k seen oe B1 for n + 1 as the numerator of a fraction 6(b) 331 cao 1

This question in 0580/41 Oct/Nov 2023

5 (a) Simplify 25x 6 2. … [2] (b) These are the first five terms of a sequence. 1 1 6 36 216 6 Find the nth term of the sequence. … [2] (c) Expand and simplify. ( x + 4)( x - 3)( 3x - 1) … [3] 7 2(d) (i) Show that ( 3x + 5) + = x simplifies to 2x + x - 3 = 0 . x - 2 [4] (ii) Solve by factorisation 2x 2 + x - 3 = 0 . x = … or x = … [3] (e) A solid cylinder has base radius x and height 3x. The total surface area of the cylinder is the same as the total surface area of a solid hemisphere of radius 5y. 2 75y 2 Show that x = . 8 [The surface area, A, of a sphere with radius r is A = 4rr 2 .] [4]

18 marks

Mark scheme: 5(a) 125x 9 final answer 2 B1 for answer 125 kx or m x 9 or for correct answer seen then spoilt 5(b) 6 n  2 oe final answer 2 B1 for answer of form 6k oe  k  1  or answer of the form   oe  6  or for correct answer seen 5(c) 3x3 + 2x2 –37x + 12 final answer 3 B2 for correct expansion of three brackets unsimplified or for simplified four-term expression of correct form with 3 terms correct or B1 for correct expansion of two brackets with at least 3 terms out of 4 correct 5(d)(i) eliminates the fraction correctly M1 eg (3x + 5) (x – 2) + 7 = x (x – 2) 3x2 + 5x –6x – 10 + 7 = x2 – 2x oe B2 B1 for 3x2 + 5x –6x –10 [ + 7] oe seen with at least 3 terms correct leading to 2x2 + x – 3 = 0 A1 dep on M1 B2 with no errors or omissions 5(d)(ii)  2 x  3  x  1 M2 or M1 for (2x + a)(x + b) where ab = −3 or 2b + a = [+]1 or for partial factors 2x(x – 1) + 3(x – 1) or x(2x + 3) –[1](2x + 3) −1.5 oe and +1 B1 5(e) 2 M1 [TSA cylinder =] 2x  2x  3 x 2 4(5 y ) 2 M1 [TSA hemisphere=] (5 y )  2 Leading to M1 dep M1M1 2x 2  6x 2  50y 2  25y 2 oe 2 75 y 2 A1 dep on M1M1M1 x  8

This question in 0580/42 May/June 2024

2 (a) The nth term of a sequence is 120 - n 3 . (i) Find the 4th term of this sequence. … [1] (ii) Find the value of n when the nth term is -1211. n = … [2] (b) The nth term of a different sequence is 3 # ( 0 .2 ) n - 1 . Find the 5th term of this sequence. … [1] (c) The table shows the first four terms of sequences A, B and C. Sequence 1st term 2nd term 3rd term 4th term 5th term nth term A 7 4 1 -2 1 2 3 4 B 4 5 6 7 C 0 2 6 12 Complete the table for each sequence. [8]

12 marks

Mark scheme: 2(a)(i) 56 1 2(a)(ii) 11 2 M1 for 120 – n3 = – 1211 or 120 – 113 = – 1211 2(b) 3 1 0.0048 or oe 625 2(c) 8 B1 for –5 A –5 10 – 3n B2 for 10 – 3n oe or B1 for k – 3n or for 10 – kn 5 B1 for 8 5 n n B B1 for oe 8 n  3 n  3 B1 for 20 2 B2 for n 2  n oe C 20 n  n or B1 for any quadratic or for at least two second differences of 2

This question in 0580/43 May/June 2024

20 These are the first five terms of a sequence. 48 24 12 6 3 (a) Find the next term. … [1] (b) Find the nth term. … [2]

3 marks

Mark scheme: 20(a) 1 3 1 1.5 or 1 or 2 2 20(b) n 2 n + k  1   1  96   oe final answer M1 for answer 96   oe  2   2   1  n or for 96   oe seen  2 

This question in 0580/41 Oct/Nov 2025

10 Find the nth term of each sequence. (a) 17, 9, 1, – 7, -15, f … [2] (b) 3, 12, 27, 48, 75, f … [2]

4 marks

Mark scheme: 10(a) 25 – 8n oe final answer 2 B1 for answer k – 8n (any k) oe or 25 – jn (j ≠ 0) or for correct answer seen then spoilt 10(b) 3n2 oe final answer 2 B1 for quadratic or for second difference = 6 (at least two, with none incorrect) or for correct answer seen then spoilt

This question in 0580/42 Oct/Nov 2025