Sequences — Paper 3 · IGCSE Mathematics 0580

9 (a) A pattern of numbers is shown below. For Examiner's row Use 1 1 2 2 3 4 3 5 6 7 8 9 4 10 11 12 13 14 15 16 5 17 18 19 20 21 22 23 24 25 6 26 … … … … … … … … … … (i) On the diagram complete row 6. [1] (ii) The last numbers in each row form a sequence. 1, 4, 9, 16, 25, …………… (a) What is the special name given to these numbers? Answer(a)(ii)(a) [1] (b) Write down the last number in the 10th row. Answer(a)(ii)(b) [1] (c) Write down an expression for the last number in the nth row. Answer(a)(ii)(c) [1] (iii) The numbers in the middle column of the pattern form a sequence. 1, 3, 7, 13, 21, 31, ………….. (a) Write down the next number in this sequence. Answer(a)(iii)(a) [1] (b) The expression for the nth number in this sequence is n2 − n + 1. Work out the 30th number. Answer(a)(iii)(b) [2] (b) Another pattern of numbers is shown below. For Examiner's Use row 1 1 2 3 4 5 6 7 8 9 10 2 11 12 13 14 15 16 17 18 19 20 3 21 22 23 24 25 26 27 28 29 30 4 31 32 33 34 35 36 37 38 39 40 (i) What is the last number in the 10th row? Answer(b)(i) [1] (ii) Find an expression for the last number in the nth row. Answer(b)(ii) [1] (iii) What is the first number in the 10th row? Answer(b)(iii) [1] (iv) Find an expression for the first number in the nth row. Answer(b)(iv) [1]

11 marks

Mark scheme: 9 a) i) 27 to 36 entered correctly 1 ii) a) square 1 b) 100 1 c) n2 c.a.o. 1 allow n x n iii) a) 43 c.a.o. 1 b) 871 2 M1 for 900 – 30 + 1 o.e. b) i) 100 1 ii) 10n c.a.o. 1 allow 10 x n iii) 91 1 vi) 10n – 9 o.e. 1 11 Total 104

This question in 0580/31 Oct/Nov 2004

8 The diagram below shows a sequence of patterns made from dots and lines. 1 dot 2 dots 3 dots 4 dots (a) Draw the next pattern in the sequence in the space above. [1] (b) Complete the table for the numbers of dots and lines. Dots 1 2 3 4 5 6 Lines 4 7 10 [2] (c) How many lines are in the pattern with 99 dots? Answer(c) [2] (d) How many lines are in the pattern with n dots? Answer(d) [2] (e) Complete the following statement. There are 85 lines in the pattern with dots. [2]

9 marks

Mark scheme: 8 (a) correct diagram (b) 13 16 19 2 1 for 2 correct (c) 298 2 M1 for evidence of a correct method (d) 3n + 1 2 1 for 3n + k (e) 28 2 M1 for evidence of a correct method [9] IGCSE – NOVEMBER 2005 0580/0581 3

This question in 0580/31 Oct/Nov 2005

1 (a) For Examiner's 2 Use 2 3 3.14 35 10 24 37 45 88 3 From the list of numbers above choose one that is (i) an irrational number, Answer(a) (i) [1] (ii) the cube root of 27, Answer(a) (ii) [1] (iii) a multiple of 9, Answer(a) (iii) [1] (iv) a prime number, Answer(a) (iv) [1] (v) a factor of 44, Answer(a) (v) [1] (vi) the product of 6 and 4. Answer(a) (vi) [1] (b) The diagram below shows a sequence of patterns made with small triangular tiles. Pattern 1 2 3 4 number (i) Draw the next pattern in the sequence. [1] (ii) Complete the table below. Pattern number 1 2 3 4 5 6 Number of tiles 1 4 9 [2] (iii) How many tiles will be in the 100th pattern? Answer(b) (iii) [1] (iv) How many tiles will be in the nth pattern? Answer(b) (iv) [1] (v) What is the special name given to the numbers in the second row of the table? Answer(b) (v) [1]

12 marks

Mark scheme: 1 (a) (i) √35 1 (ii) 3 1 (iii) 45 1 (iv) 2 or 3 or 37 1 accept any combination (v) 2 1 (vi) 24 1 (b) (i) Correct arrangement of triangles drawn. 1 accept if only 1 internal line missing (ii) 16 25 36 2 1 mark for 2 correct (iii) 10000 or 1 x 104 1 Not 1002 (iv) n2 or n × n 1 accept t = n2 etc. do not accept x2 (v) Square (numbers) 1 accept squares, squared 12

This question in 0580/31 Oct/Nov 2006

9 In the pattern below each diagram shows a letter E formed by joining dots. For Examiner's Diagram 1 Diagram 2 Diagram 3 Diagram 4 Use (a) Draw the next letter E in the pattern. [1] (b) Complete the table showing the number of dots in each letter E. Diagram 1 2 3 4 5 Dots 8 15 [3] (c) How many dots make up the letter E in (i) Diagram 10, Answer(c)(i) [2] (ii) Diagram n? Answer(c)(ii) [2] (d) The letter E in Diagram n has 113 dots. Write down an equation in n and use it to find the value of n. Answer(d) n = [3]

11 marks

Mark scheme: 9 (a) Letter E correctly drawn B1 (b) 22, 29, 36 B3 B1 for each correct number. (c) (i) 71 B2 B1 for 7 × 10 + 1 or 8 + 9 × 7 seen. (ii) 7n + 1 or 8 + (n – 1) × 7 oe B2 SC1 for 7n + k seen. (k is an integer) oe (d) Their (c)(ii) = 113 B1ft ft any expression involving n. Full method of solution of their M1ft ft only a linear equation. equation. (113 – k)/ ‘7’ 16 A1cao www B2 [11]

This question in 0580/31 May/June 2007

10 For Examiner's Use Diagram 1 Diagram 2 Diagram 3 Diagram 4 Diagram 5 Look at the sequence of five diagrams above. Diagram 1 has 2 dots and 1 line. Diagram 2 has 6 dots and 7 lines. The numbers of dots and lines in each of the diagrams are shown in the table below. Diagram number 1 2 3 4 5 6 7 Number of dots 2 6 12 20 30 Number of lines 1 7 17 31 49 (a) Fill in the empty spaces in the table for Diagrams 6 and 7. [4] (b) How many dots are there in Diagram n? Answer(b) [2] (c) The number of lines in Diagram n is 2n2 – 1. Which diagram has 287 lines? Answer(c) [2]

8 marks

Mark scheme: 10 (a) 42, 56 B1B1 cao 71, 97 B1B1 cao (b) n (n + 1) oe B2 M1 for attempt at length x width involving n or n'th (n'th + 1) or k (k + 1) where k is any variable (c) 12 B2 M1 for 2 n² – 1 = 287 [8]

This question in 0580/31 Oct/Nov 2007

10 The first three diagrams in a sequence are shown below. For Each diagram has one more trapezium added on the right. Examiner's Use Diagram 1 Diagram 2 Diagram 3 (a) Complete the table which shows the number of lines and dots in each diagram. Diagram 1 2 3 4 Number of lines 4 7 Number of dots 4 6 [2] (b) Find the number of lines and dots in Diagram 10. Answer(b) lines and dots [2] (c) For Diagram n, write down in terms of n, the number of (i) lines, Answer(c)(i) [2] (ii) dots. Answer(c)(ii) [2] (d) Find the difference, in terms of n, between your answers to parts (c)(i) and (c)(ii). Simplify your answer. Answer(d) [2]

10 marks

Mark scheme: 10 (a) (Lines) 10 and 13 W1 (Dots) 8 and 10 W1 (b) (Lines) 31, (Dots) 22 W1, W1 (c) (i) 3n + 1 oe SC1 for jn + 1 or 3n + k W2cao where j and k are integers. j ≠ 0 (ii) 2n + 2 oe SC1 for jn + 2 or 2n + k W2cao where j and k are integers. j ≠ 0 (d) n − 1 or 1 − n M1 for ‘(3n + 1)’ − ‘(2n + 2)’ or reversed W2ft Ft and M1 dependent on two linear algebraic expressions

This question in 0580/31 Oct/Nov 2008

9 (a) The first four terms of a sequence are 12, 7, 2, –3. For Examiner's (i) Write down the next two terms of the sequence. Use Answer(a)(i) and [2] (ii) State the rule for finding the next term of the sequence. Answer(a)(ii) [1] (iii) Write down an expression for the nth term of this sequence. Answer(a)(iii) [2] (b) The first four terms of another sequence are −3, 2, 7, 12. Write down an expression for the nth term of this sequence. Answer(b) [2] (c) Add together the expressions for the nth terms of both sequences. Write your answer as simply as possible. Answer(c) [1]

8 marks

Mark scheme: 9 (a) (i) −8, 1cao −13 1ft Ft sixth term 5 less than the fifth (ii) Subtract 5 oe 1 (iii) −5n + 17 2 W1 for jn + 17 or –5n + k where j and k are integers, j ≠ 0 (b) 5n − 8 2 W1 for jn − 8 or 5n – k where j and k are integers, j ≠ 0 (c) 1ft Ft two linear expressions only 9 www

This question in 0580/31 May/June 2009

10 For Examiner's Use Diagram 1 Diagram 2 Diagram 3 Diagram 4 Diagram 5 Each of the diagrams above shows one small shaded square and a number of small unshaded squares. The diagrams form a sequence. (a) Complete Diagram 5. [1] (b) Complete the table. Diagram 1 2 3 4 5 50 n Total number of 1 4 9 16 small squares Number of small 1 1 1 1 shaded squares Number of small 0 3 8 15 unshaded squares [7] (c) Diagram p has 9999 small unshaded squares. Find p. Answer(c) p = [1]

9 marks

Mark scheme: 10 (a) 5 by 5 shape 1 (b) First row 25 2500 n² 1, 1, 1 Independent Second row 1 1 1 1 All three Third row 24 2499 n² – 1 1, 1, 1 Independent (c) 100 1

This question in 0580/31 Oct/Nov 2010

9 For B D Examiner's Use 2.7 cm NOT TO cm 2.7 SCALE C A E (a) In the diagram above, AB and ED are vertical. The diagram is symmetrical about a line through C parallel to AB. Angle BCD = 90° and BC = CD = 2.7 cm. (i) Calculate BD. Answer(a)(i) BD = cm [2] (ii) Complete the statement. Triangle BCD is right-angled and [1] (iii) Find the size of angle ABC. Answer(a)(iii) Angle ABC = [1] For Examiner's Use Diagram 1 Diagram 2 Diagram 3 Diagram 4 (b) The pattern of diagrams above is continued by adding more lines and dots. (i) On the grid, draw diagram 4. [1] (ii) Complete the table below. Diagram 1 2 3 4 5 Number of lines 4 7 [2] (c) How many lines will there be in (i) Diagram 9, Answer(c)(i) [1] (ii) Diagram n? Answer(c)(ii) [2] (d) The number of lines in Diagram r is 76. Find the value of r. Answer(d) r = [2] (e) Write down an expression, in terms of n, for the number of dots in Diagram n. Answer(e) [1]

13 marks

Mark scheme: 9 (a) (i) 3.82 art 2 M1 for 2.72 + 2.72 or better 27 or sin 45 = or better BD 27 or cos 45 = or better BD (ii) Isosceles 1 (iii) 45 cao 1 (b) (i) Diagram 4 1 (ii) 10, 13, 16 2 B1 for 2 correct or difference of 3 seen between diagram 4 and diagram 5 in table (c) (i) 28 1 (ii) 3n + 1 oe 2 B1 for pn + 1 (p ≠ 0) or 3n + q (d) 25 2ft M1 for 76 = their (c)(ii) (if linear) (e) 3n + 2 oe 1ft ft their (c)(ii) + 1 (must be a linear expression)

This question in 0580/32 Oct/Nov 2010

11 (a) (i) For Examiner's 0, 1, 1, 2, 3, 5, 8, …. Use This sequence has the rule: After the first two terms, any term is the sum of the two previous terms. The first two terms are 0 and 1, the 3rd term is 0 + 1 = 1, the 4th term is 1 + 1 = 2, the 5th term is 1 + 2 = 3 and so on. Show that the 8th term is 13. Answer(a)(i) [1] (ii) Each of the following sequences have the same rule as part (a)(i). For each sequence write down the missing terms. 2, 5, 7, , [1] 4, 3, 7, , [1] 5, 2, , [1] 0, , 3, [1] 1, , , 9, [1] , , 5, 7 [1] (b) For the following sequences find the next term and the n th term. (i) 1, 3, 5, 7, 9, n th term = [3] (ii) 1, 4, 9, 16, 25, n th term = [2] 1 1 1 1 (iii) 1, , , , , n th term = [2] 2 3 4 5

14 marks

Mark scheme: 11 (a) (i) 5 + 8 (= 13) 1 (ii) 12, 19 1 10, 17 1 7, 9 1 3, 6 1 4, 5 1 3, 2 1 (b) (i) 11 1 2n – 1 2 B1 for 2n ± k or jn – 1 (j ≠ 0) (ii) 36 n2 1, 1 1 1 (iii) 1, 1 6 n

This question in 0580/33 Oct/Nov 2010

9 For Examiner's Use Diagram 1 Diagram 2 Diagram 3 Diagram 4 Diagram 5 The Diagrams above form a pattern. (a) Draw Diagram 5 in the space provided. [1] (b) The table shows the numbers of dots in some of the diagrams. Complete the table. Diagram 1 2 3 4 5 10 n Number of dots 3 5 [5] (c) What is the value of n when the number of dots is 737? Answer(c) [2] (d) Complete the table which shows the total number of dots in consecutive pairs of diagrams. For example, the total number of dots in Diagram 2 and Diagram 3 is 12. Diagrams 1 and 2 2 and 3 3 and 4 4 and 5 10 and 11 n and n + 1 Total number of 8 12 16 dots [3]

11 marks

Mark scheme: 9 (a) Diagram drawn 1 (b) 7, 9, 11 2 B1 for 2 correct 21 1 2n + 1 oe 2 SC1 for 2n + or – any integer (c) 368 2ft Must be integer for 2 marks M1 for their 2n + 1 = 737 ft if linear (d) 20, 44, 1, 1 4(n + 1) oe 1

This question in 0580/33 May/June 2011

11 For Examiner's Use Diagram 1 Diagram 2 Diagram 3 The diagrams show a sequence of shapes. (a) On the grid, draw Diagram 4. [1] (b) Complete the table showing the number of lines in each diagram. Diagram (n) Number of lines 1 6 2 11 3 4 5 [3] (c) Work out the number of lines in Diagram 8. Answer(c) [1] (d) Write down an expression, in terms of n, for the number of lines in Diagram n. Answer(d) [2] (e) Work out the number of lines in Diagram 100. Answer(e) [1] (f) The number of lines in Diagram p is 66. Find the value of p. Answer(f) p = [2]

10 marks

Mark scheme: 11 (a) Correct shape drawn 1 (b) 16, 21, 26 3 B1 for each SC1 “their 16” + 5 SC1 “their 21” + 5 (c) 41 1 (d) 5n + 1 2 B1 for 5n, B1 for +1 (e) 501 1ft Their (d) if linear (f) 13 2ft Their (d) if linear B1 for their (d) = 66

This question in 0580/31 Oct/Nov 2011

10 (a) Write down the next term in each of the following sequences. For Examiner's Use (i) 2, 9, 16, 23, [1] (ii) 75, 67, 59, 51, [1] (iii) 2, 5, 9, 14, [1] 1 1 (iv) 2, 1, , , [1] 2 4 (v) 2, 4, 8, 16, [1] (b) For the sequence in part (a)(i) write down (i) the 10th term, Answer(b)(i) [1] (ii) the nth term. Answer(b)(ii) [2] n 2 + 3n (c) The nth term of the sequence in part (a)(iii) is . 2 Calculate the 50th term of this sequence. Answer(c) [2] (d) The nth term of the sequence in part (a)(v) is 2n. Calculate the 12th term of this sequence. Answer(d) [1]

11 marks

Mark scheme: 10 (a) (i) 30 1 (ii) 43 1 (iii) 20 1 1 (iv) or 0.125 1 8 (v) 32 1 (a) (i) 65 1 (ii) 7n − 5 or equivalent 2 B1 for 7n seen 502 + 3 × 50 (c) 1325 2 B1 for or better seen 2 (d) 4096 1

This question in 0580/32 Oct/Nov 2011

11 (a) Write down the next term in each of the following sequences. For Examiner's Use (i) 8, 15, 22, 29, [1] (ii) 3, 6, 12, 24, [1] (iii) 1, 4, 9, 16, [1] (iv) 0, 3, 8, 15, [1] (b) Write down an expression, in terms of n, for the nth term of (i) the sequence in part(a)(iii), Answer(b)(i) [1] (ii) the sequence in part(a)(iv). Answer(b)(ii) [1] (c) The nth term of a sequence is 7n –3 . (i) Write down the value of the 4th term. Answer(c)(i) [1] (ii) Which term has a value of 592? Answer(c)(ii) [2] (d) 1, 2, 2, 4, 8, 32, 256, …… Work out the next two terms of this sequence. Answer(d) , [2]

11 marks

Mark scheme: 11 (a) 36, 48, 25, 24 ft 4 B1 each ft their 25 – 1 (b) (i) n2 oe 1 (ii) n2 – 1 oe 1ft ft their (i) – 1, if expression in n (c) (i) 25 1 (ii) 85 2 M1 for 7n – 3 = 592 or better (d) 8192, 2 097 152 2 B1 each SC1ft 256 × their 8192

This question in 0580/33 Oct/Nov 2011

5 For Examiner's Use Diagram 1 Diagram 2 Diagram 3 Diagram 4 The number of crosses in each Diagram forms a sequence. (a) On the grid draw Diagram 4. [1] (b) Write down the number of crosses needed to draw Diagram 5. Answer(b) [1] (c) Diagram 1 has 1 row of 3 crosses. Diagram 2 has 2 rows of 4 crosses. (i) Complete this statement for Diagram n. Diagram n has n rows of crosses. [1] (ii) Write down, in terms of n, how many crosses are needed to draw Diagram n. Answer(c)(ii) [1] (iii) Find the number of crosses needed to draw Diagram 20. Answer(c)(iii) [1]

5 marks

Mark scheme: their (a)(i) (b) 5 1ft Ft is 34 (c) Said by 1.5 secs 3ft ' their (a)(ii)' M1ft (= 32.5) 4 ' their (a)(ii)' M1ft 34 – (34 – 32.5) 4 120 120 (d) (i) 67.4° 2 M1 ‘tan’= or ‘sin’= 50 their 130 50 or ‘cos’= their 130 (ii) 113° or 112.6° 1ft 180 – ‘their (d)(i)’

This question in 0580/31 May/June 2012

10 The Patterns shown below form a sequence. For Examiner's Pattern 1 has 6 dots and 6 lines. Use Pattern 2 has 10 dots and 11 lines. Pattern 1 Pattern 2 Pattern 3 Pattern 4 (a) On the grid, draw Pattern 4. [1] (b) (i) Find the number of dots in Pattern 5. Answer(b)(i) [1] (ii) Explain how you worked out your answer in part (b)(i). Answer(b)(ii) [1] (c) Write down an expression, in terms of n, for the number of dots in Pattern n. Answer(c) [2] (d) The number of dots in Pattern n is 62 . Find n. Answer(d) n = [2]

7 marks

Mark scheme: 10 (a) correct pattern 1 (b) (i) 22 1 (ii) add 4 1 must have 4 with a direction, accept plus 4 (c) 4n + 2 or 4(n – 1) + 6 oe 2 B1 for 4n + j or kn + 2 (k ≠ 0) seen (d) 15 cao 2 M1 their (c) = 62 or multiple additions or subtractions

This question in 0580/33 May/June 2012

6 For Examiner's Use Diagram 1 Diagram 2 Diagram 3 Diagram 4 A sequence of diagrams is made from black counters and white counters. The first four diagrams in the sequence are shown. (a) Complete the table. Diagram 1 2 3 4 5 Number of black counters 1 4 Number of white counters 1 4 [4] (b) Complete the statement. The numbers of black counters are all numbers. [1] (c) How many white counters are needed for (i) Diagram 8, Answer(c)(i) [1] (ii) Diagram n? Answer(c)(ii) [2] (d) Diagram p contains 58 white counters. For Examiner's Use (i) Find the value of p. Answer(d)(i) p = [2] (ii) Find the number of black counters in Diagram p. Answer(d)(ii) [1]

11 marks

Mark scheme: 6 (a) 9 16 25 2 B1 for 2 correct 7 10 13 2 B1 for 2 correct, or difference of 3 between diagrams 4 and 5 (b) square 1 (c) (i) 22 1 (ii) 3n – 2 oe final answer 2 B1 for 3n ± j seen Or kn – 2, where k ≠ 0 (d) (i) 20 2 ft M1 for ‘their (c)(ii)’ = 58 or better, seen (ii) 400 1ft ‘their (d)(i)’2 (must be evaluated) IGCSE – October/November 2012 0580 31

This question in 0580/31 Oct/Nov 2012

8 For Examiner's Use Diagram 1 Diagram 2 Diagram 3 Diagram 4 (a) This pattern of diagrams forms a sequence. (i) On the grid, draw Diagram 4. [1] (ii) Complete this table. Diagram 1 2 3 4 5 Number of dots 7 12 [2] (b) How many dots will there be in (i) Diagram n, Answer(b)(i) [2] (ii) Diagram 29. Answer(b)(ii) [1] (c) There are either 2 lines or 3 lines meeting at the dots in the Diagrams. For Examiner's Use In Diagram 1 there are 0 dots where 3 lines meet. In Diagram 2 there are 4 dots where 3 lines meet. Complete the statements. (i) In Diagram 3 there are dots where 3 lines meet. [1] (ii) In Diagram n there are dots where 3 lines meet. [2] (d) Find the number of dots where 2 lines meet in Diagram n. Answer(d) [1] Question 9 is printed on the next page.

10 marks

Mark scheme: 8 (a) (i) Diagram 4 correctly drawn 1 Clear intention (ii) 17 22 27 2 B1 for 2 correct or a gap of 5 between Diagrams 3 and 4 and 4 and 5. (b) (i) 5n + 2 oe final answer 2 B1 for jn + 2 (j ≠ 0) or 5n + k (ii) 147 1ft Ft a linear expression (c) (i) 8 1 (ii) 4n − 4 oe final answer 2 B1 for jn − 4 (j ≠ 0) or 4n + k (d) n + 6 cao 1

This question in 0580/32 Oct/Nov 2012

6 (a) These are the first four terms of a sequence. For Examiner's Use 19 15 11 7 (i) Write down the next two terms of this sequence. Answer(a)(i) and [2] (ii) Write down the rule for finding the next term of this sequence. Answer(a)(ii) [1] (iii) Find an expression for the nth term of this sequence. Answer(a)(iii) [2] (b) The nth term of another sequence is 2n + 6 . Write down the first three terms of this sequence. Answer(b) , , [2] (c) The first three diagrams of a different sequence are shown below. Diagram 1 Diagram 2 Diagram 3 Complete the table. Diagram 1 2 3 8 n Number of lines 6 9 12 [3]

10 marks

Mark scheme: 6 (a) (i) 3 − 1 1,1 If B0 award B1 if term 2 − term 1 = − 4 (ii) subtract 4 1 Accept minus 4, take away 4 (iii) − 4n + 23 oe final answer 2 M1 −4n+ k or jn+23 ( j ≠ 0) as answer (b) 8, 10, 12 2 M1 2 correct terms SC1 for 6, 8, 10 (c) 27, 3n+ 3 oe final answer 3 B1 27 B1 3n + k or jn + 3 ( j ≠ 0) IGCSE – October/November 2012 0580 33

This question in 0580/33 Oct/Nov 2012

10 (a) (i) Find the highest common factor (HCF) of 24 and 36. For Examiner′s Use Answer(a)(i) … [2] (ii) Factorise. 24x + 36y Answer(a)(ii) … [1] (b) Simplify. (i) w + 8k – 5w + 2k Answer(b)(i) … [2] (ii) (x4)5 Answer(b)(ii) … [1] (c) Here are the fi rst four terms of a sequence. 7 11 15 19 Find the nth term of this sequence. Answer(c) … [2] (d) Solve the simultaneous equations. 3x + y = 8 x + 5y = 5 Answer(d) x = … y = … [3]

11 marks

Mark scheme: 10 (a) (i) 12 2 B1 for any other common factor other than 1 (ii) 12(2x + 3y) cao 1 (b) (i) 10k – 4w 2 B1 for either 10k ± nw or qk – 4w p,q ≠ 0 (ii) x20 1 (c) 4n + 3 oe final answer 2 B1 for 4n + c or kn + 3 , k ≠ 0 (d) [x] = 2.5, [y] = 0.5 3 M1 for correct method to eliminate one variable. A1 for x or y correct.

This question in 0580/32 May/June 2013

9 (a) Write down the next term and the rule for fi nding the next term for the following sequences. For Examiner′s Use (i) 3, 9, 27, 81, ... Answer(a)(i) Next term … rule … [2] (ii) 2, 3, 6, 11, 18, ... Answer(a)(ii) Next term … rule … [2] 1 (iii) 4, 2, 1, , ... 2 Answer(a)(iii) Next term … rule … [2] (iv) 5, –10, 20, –40, ... Answer(a)(iv) Next term … rule … [2] (b) (i) Write down the next two terms of this sequence. 5, 13, 21, 29, … , … [2] (ii) Write down the nth term of this sequence. Answer(b)(ii) … [2] (iii) Find the 100th term. Answer(b)(iii) … [1] _____________________________________________________________________________________

13 marks

Mark scheme: 9 (a) (i) 243 1 Multiply by 3 oe 1 (ii) 27 1 Add next odd number oe 1 Add 1 first and keep adding 2 more each time (iii) 1 or 0.25 1 4 Halve or divide by 2 1 (iv) 80 1 Multiply by –2 oe 1 (b) (i) 37, 45 1, 1ft ft is (ans) + 8 (ii) 8n – 3 oe final answer 2 B1 for 8n + a or B1 for bn – 3 (b ≠ 0) (iii) 797 1ft Only follow through a linear expression

This question in 0580/33 May/June 2013

6 (a) Here are three different sequences. For Examiner′s Write the missing terms in the spaces provided. Use (i) 2, 8, 14, 20, … [1] (ii) 1, 4, 9, … , 25 [1] (iii) … , 12, 7, 2, … [2] (b) Here is the rule for fi nding the next term in another sequence. Double the previous term and subtract 1. The fi rst two terms in this sequence are 3 and 5. (i) Work out the next two terms in the sequence. Answer(b)(i) … , … [2] (ii) Complete the following statement. All the terms in this sequence are … numbers. [1] (c) Here is the start of a sequence of stick patterns. Pattern 1 Pattern 2 Pattern 3 8 sticks 13 sticks 18 sticks (i) Find the number of sticks in Pattern 4. Answer(c)(i) … [1] (ii) Write down an expression for the number of sticks in Pattern n. Answer(c)(ii) … [2] (iii) One pattern in the sequence has 98 sticks. Which pattern number is this? Answer(c)(iii) … [2] _____________________________________________________________________________________

12 marks

Mark scheme: 6 (a) (i) 26 1 (ii) 16 1 (iii) 17 –3 2 B1 for each (b) (i) 9 17 2 B1 for one correct in correct position or FT for fourth term (ii) odd 1 (c) (i) 23 1 (ii) 5n + 3 oe final answer 2 B1 for 5n + k , jn + 3 j ≠ 0 Or 5n + 3 oe not as final answer (iii) 19 2 M1FT for their (c)(ii) = 98 if linear soi

This question in 0580/32 Oct/Nov 2013

8 Here is a sequence of patterns made using identical polygons. For Examiner′s Use Pattern 1 Pattern 2 Pattern 3 (a) Write down the mathematical name of the polygon in Pattern 1. Answer(a) … [1] (b) Complete the table for the number of vertices (corners) and the number of lines in Pattern 3, Pattern 4 and Pattern 7. Pattern 1 2 3 4 7 Number of vertices 8 14 Number of lines 8 15 [5] (c) (i) Find an expression for the number of vertices in Pattern n. Answer(c)(i) … [2] (ii) Work out the number of vertices in Pattern 23. Answer(c)(ii) … [1] (d) Find an expression for the number of lines in Pattern n. For Examiner′s Use Answer(d) … [2] (e) Work out an expression, in its simplest form, for (number of lines in Pattern n) – (number of vertices in Pattern n). Answer(e) … [2] _____________________________________________________________________________________ Question 9 is printed on the next page.

13 marks

Mark scheme: 8 (a) Octagon 1 (b) [Pattern 3] 20 and 22 1 [Pattern 4] 26, 29 1, 1 [Pattern 7] 44, 50 1, 1 (c) (i) 6n + 2 oe final answer 2 B1 for 6n + a or bn + 2 b ≠ 0 (ii) 140 oe 1FT ft linear expression in (c)(i) (d) 7n + 1 oe final answer 2 B1 for 7n + c or dn + 1 d ≠ 0 (e) n – 1 final answer 2FT B1FT for n + j or kn 1 k ≠ 0 3V 2 3V 3V

This question in 0580/33 Oct/Nov 2013

9 (a) For these sequences, write down the next two terms and the rule for fi nding the next term. (i) 84, 75, 66, 57, . . . Answer(a)(i) … , … rule … [3] (ii) 2, 6, 18, 54, . . . Answer(a)(ii) … , … rule … [3] (b) For the sequence in part (a)(i), (i) write down an expression, in terms of n, for the n th term, Answer(b)(i) … [2] (ii) fi nd the 21st term. Answer(b)(ii) … [2] __________________________________________________________________________________________

10 marks

Mark scheme: 9 (a) (i) 48, 39 1, 1FT FT 6th term = 5th term −9 Subtract 9 oe 1 (ii) 162, 486 1, 1FT FT 6th term = 5th term × 3 Multiply by 3 oe 1 (b) (i) 93 – 9n oe final answer 2 B1 for −9n + c or kn + 93, k ≠ 0 (ii) −96 cao 2 M1 for substitution of n = 21 into their linear expression

This question in 0580/31 May/June 2014

9 Diagram 1 Diagram 2 Diagram 3 Diagram 4 Diagrams 1 to 4 show a sequence of shapes made up of lines and dots at the intersections of lines. (a) (i) Complete the table showing the number of dots in each diagram. Diagram 1 2 3 4 5 6 Dots 3 8 13 [3] (ii) Write down the rule for continuing the sequence of dots. Answer(a)(ii) … [1] (iii) Write down an expression, in terms of n, for the number of dots in Diagram n. Answer(a)(iii) … [2] (iv) Find the number of dots in Diagram 15. Answer(a)(iv) … [1] (b) The dots are joined by sloping lines and horizontal lines. (i) Diagram 1 has 2 sloping lines and Diagram 2 has 6 sloping lines. Find the number of sloping lines in Diagrams 3 and 4. Answer(b)(i) Diagram 3 … Diagram 4 … [2] (ii) Write down an expression, in terms of n, for the number of sloping lines in Diagram n. Answer(b)(ii) … [2]

11 marks

Mark scheme: 9 (a) (i) 18 23 28 1, 1, 1 Allow one mark for each addition of 5 to the previous answer (ii) Add 5 oe 1 (iii) 5n – 2 oe 2 B1 for 5n + j or kn – 2 k ≠ 0 (iv) 73 1FT FT their (a)(iii) if linear. (b) (i) 10 14 1, 1 Allow 1 mark for addition of 4 on their value for 3rd diagram. (ii) 4n – 2 oe 2 B1 for 4n + j or kn – 2 k ≠ 0

This question in 0580/31 Oct/Nov 2014

2 (a) Write down the mathematical name of a polygon with 8 sides. Answer(a) … [1] (b) Calculate the interior angle of a regular 8-sided polygon. Answer(b) … [3] (c) Diagram 1 Diagram 2 Diagram 3 The pattern of diagrams above forms a sequence. (i) Complete the table. Diagram 1 2 3 4 5 Number of dots 8 15 [2] (ii) Find an expression, in terms of n, for the number of dots in Diagram n. Answer(c)(ii) … [2] (iii) Find the number of dots in Diagram 10. Answer(c)(iii) … [1] (iv) Find the value of n for a diagram with 92 dots. Answer(c)(iv) … [2] __________________________________________________________________________________________

11 marks

Mark scheme: 2 (a) Octagon 1 (b) 135 3 M2 for 180 – (360 ÷ 8) or M2 for (8 − 2) × 180 8 or M1 for (360 ÷ 8) or M1 for (8 – 2) × 180 (c) (i) 22 29 36 2 B1 for two terms in correct places or 2 terms with a difference of 7. (ii) 7n + 1 oe 2 B1 for 7n + j or kn + 1 (k ≠ 0) (iii) 71 1FT FT for their (c)(ii) if linear (iv) 13 nfww 2 M1FT for their (c)(ii) = 92 or M1 for (92 – 1) ÷ 7 or 91 ÷ 7 or M1 for 7 × 13 + 1 = 92

This question in 0580/32 Oct/Nov 2014

8 (a) A pattern of calculations is shown below. Complete the last four rows. 32 – 12 = 4 × 2 42 – 22 = 4 × 3 52 – 32 = 4 × 4 62 – … = … × 102 – … = … × … – … = 4 × 100 … – … = 4 × n [5] (b) These are the first five terms in a sequence. 3 7 11 15 19 (i) Write down the next term in the sequence. Answer(b)(i) … [1] (ii) Write down an expression for the nth term. Answer(b)(ii) … [2] (iii) Work out the 57th term. Answer(b)(iii) … [1] (iv) Is the number 237 a term in the sequence? Give a reason for your answer. Answer(b)(iv) … because … … [2] __________________________________________________________________________________________ Question 9 is printed on the next page.

11 marks

Mark scheme: 8 (a) 42, 4 × 5 1 82, 4 × 9 1 1012, 992 1 (n + 1)2, (n – 1)2 2 SC1 for (n + 1)2 or (n – 1)2 seen or for n + 12 and n – 12 (b) (i) 23 1 (ii) 4n – 1 oe 2 M1 for 4n seen (iii) 227 1FT FT from (b)(ii) if in form jn + k j,k ≠ 0 (iv) No, oe, with valid reason 2 M1FT for (227), (231), 235 or ft from their their (b)(ii) − k (b)(iii) or 59.5 or ft j A1 for correct deduction and mention of 237 between 235 and 239 or 59.5 is not a whole number oe

This question in 0580/32 Feb/March 2015

9 (a) Here are the first four terms of a sequence. 5 8 11 14 (i) Write down the next term in this sequence. Answer(a)(i) … [1] (ii) Write down the rule for finding the next term of this sequence. Answer(a)(ii) … [1] (iii) Find an expression for the nth term of this sequence. Answer(a)(iii) … [2] (iv) Explain why the number 300 is not in this sequence. Answer(a)(iv) … [1] (b) Here are the first four terms of another sequence. 4 7 11 16 (i) Write down the next two terms in this sequence. Answer(b)(i) … , … [2] (ii) Write down the rule for continuing this sequence. Answer(b)(ii) … [1]

8 marks

Mark scheme: 9 (a) (i) 17 1 (ii) add 3 or +3 1 (iii) 3n + 2 oe as final answer 2 B1 for 3n + k or jn + 2 ( j ≠ 0) (iv) 300 is in the 3 times table [and all 1 accept any correct reason the terms are 1 less or 2 more than the 3 times table] (b) (i) 22 29 2 B1 for either correct or SC1 for a difference between the two terms of 7 (ii) the difference increases by one each 1 accept any correct explanation time

This question in 0580/33 May/June 2015

2 Here are the first four diagrams in a sequence. Diagram 1 Diagram 2 Diagram 3 Diagram 4 Diagram 5 (a) On the grid, draw Diagram 5. [1] (b) Complete the table below for Diagram 4 and Diagram 5. Diagram Number of Number of Total number number s s of s and s 1 1 0 1 2 3 1 4 3 6 3 9 4 5 [2] (c) Find an expression, in terms of n, for the total number of s and s in Diagram n. Answer(c) … [1] (d) Find the total number of s and s in Diagram 23. Answer(d) … [1] (e) Describe in words the rule for continuing the sequence for the number of s. 1, 3, 6, … Answer(e) … [1]

6 marks

Mark scheme: 2 (a) 1 O X X X X O O X X X O O O X X O O O O X O O O O O (b) 10, 6, 16 2 M1 for 4 or 5 correct numbers or for one 15, 10, 25 correct row (c) n2 1 (d) 529 1FT FT their (c) if algebraic expression (e) Add on 2, then 3, then 4 etc. oe 1

This question in 0580/32 Oct/Nov 2015

10 The first three diagrams in a sequence are shown below. Diagram 1 Diagram 2 Diagram 3 (a) Complete the table for the number of lines and the number of dots in Diagram 3 and Diagram 4. Diagram 1 2 3 4 Lines 5 10 Dots 6 11 [2] (b) For Diagram n, write down an expression, in terms of n, for the number of (i) lines, Answer(b)(i) … [1] (ii) dots. Answer(b)(ii) … [1] (c) Work out the number of lines and the number of dots in Diagram 20. Answer(c) Number of lines = … Number of dots = … [2] __________________________________________________________________________________________

6 marks

Mark scheme: 10 (a) 15 20 2 B1 for 1 correct row or column 16 21 (b) (i) 5n oe final answer 1 (ii) 5n + 1 oe final answer 1 FT FT algebraic expression (c) 100 1 101 1

This question in 0580/33 Oct/Nov 2015

9 (a) A solid has 6 faces, 8 vertices and 12 edges. All the edges have the same length. Write down the mathematical name of this solid. … [1] (b) Here is a sequence of diagrams made from identical square tiles. Diagram 1 Diagram 2 Diagram 3 Diagram 4 (i) On the grid, draw Diagram 4. [1] (ii) Complete the table. Diagram 1 2 3 4 5 Number of tiles 1 5 9 [2] (iii) Find an expression, in terms of n, for the number of tiles in Diagram n. … [2] (iv) Find the number of tiles in Diagram 19. … [1] (v) A box contains 98 of these tiles. (a) Diagram x is made from as many tiles as possible from this box. Find the value of x. x = … [2] (b) When Diagram x is made, how many tiles are left in the box?

9 marks

Mark scheme: 9 (a) Cube 1 (b) (i) 1 (ii) 13 1 17 1 If 0 scored SC1 for second number 4 more than the first (iii) 4n – 3 oe final answer 2 B1 for 4n – j or kn – 3 ( k ≠ 0 ) (iv) 73 1FT follow through linear expressions in (b)(iii) (v)(a) 25 2 B1FT for their (b)(iii) = 98 or B1 for 25.25 (v)(b) 1 1FT follow through their (b)(v)(a) if an integer

This question in 0580/33 May/June 2016

6 (a) Here are the first four terms of a sequence. 18 25 32 39 (i) Write down the next term. … [1] (ii) Explain how you worked out your answer. … [1] (b) The nth term of another sequence is n2 + 3. Write down the first three terms of this sequence. … , … , … [2] (c) Simplify. (i) 6a + 5h − 4a − 8h … [2] (ii) 5(x + 3) + 4(2x − 6) … [2] (d) Factorise. 6g + 15 … [1] (e) A rectangle has length (x + 6) cm and width 5 cm. The area of this rectangle is 85 cm2. Find the value of x. x = … [3]

12 marks

Mark scheme: 6 (a) (i) 46 1 (ii) Add 7 oe 1 (b) 4, 7, 12 2 M1 for 2 correct or 3 , 4 , 7 (c) (i) 2a – 3h final answer 2 B1 for 2a or −3h (ii) 13x – 9 final answer 2 M1 for 5x + 15 or 8x – 24 or 13x or −9 (d) 3( 2g + 5) final answer 1 (e) 11 nfww 3 M2 for 5x = 55 or x + 6 = 17 or M1 for 5x + 30 [ = 85] or 5 (x + 6 ) [ = 85] or M1 for correct first step of incorrect linear equation if of the form ax + b = 85, a ≠ 1

This question in 0580/31 Oct/Nov 2016

9 A sequence of patterns is made from lines and dots. The first three patterns in the sequence are shown. Pattern 1 Pattern 2 Pattern 3 Pattern 4 (a) Draw Pattern 4 on the grid. [1] (b) Complete the table. Pattern 1 2 3 4 10 Number of dots 2 3 Number of lines 4 7 [4] (c) Find an expression, in terms of n, for (i) the number of dots in Pattern n, … [1] (ii) the number of lines in Pattern n. … [2] (d) One of these patterns has 76 lines. Work out how many dots are in this pattern. … [2]

10 marks

Mark scheme: 9 (a) 1 (b) 4 5 11 4 B1 for 11 10 13 31 B1 for 31 B2 for 4, 5, 10, 13 or B1 for two of 4, 5, 10, 13 (c) (i) n + 1 oe final answer 1 (ii) 3n + 1 oe final answer 2 B1 for 3n + k or cn + 1 c≠0 (d) 26 2 M1FT for their c(ii) = 76 or better or M1 implied by answer of 25

This question in 0580/32 Oct/Nov 2016

7 Here is a sequence of diagrams made using identical rectangles. A dot is shown at the junction of three lines. A cross is shown at the junction of two lines. x x x x x x x x x x x x x x x x x x x x x x x x Diagram 1 Diagram 2 Diagram 3 Diagram 4 (a) Write down the order of rotational symmetry of Diagram 1. … [1] (b) Complete Diagram 4 using dots and crosses. [1] (c) Complete the table for Diagram 4 and Diagram 5. Diagram 1 2 3 4 5 Number of dots 0 4 10 Number of crosses 4 6 8 [3] (d) (i) Describe, in words, the rule for continuing the sequence for the number of dots. … [1] (ii) The expression for the number of dots in Diagram n is n 2 + n - 2 . Find the number of dots in Diagram 12. … [2] (e) (i) Write down an expression for the number of crosses in Diagram n. … [2] (ii) Diagram n has 100 crosses. Find the value of n. n = … [2]

12 marks

Mark scheme: 7(a) 2 1 7(b) 3 dots correctly placed 1 4 crosses correctly placed 7(c) 18 28 1,1 If zero scored, SC1 for their 18 + 10 10 12 1 7(d)(i) Add two more each time oe 1 7(d)(ii) 154 2 M1 for 122 + 12 − 2 7(e)(i) 2n + 2 oe final answer 2 B1 for 2n + j or kn + 2 (k ≠ 0 or 1) 7(e)(ii) 49 2 M1 for their (e)(i) = 100 provided (e)(i) is algebraic soi

This question in 0580/31 May/June 2017

9 (a) Write down the next two terms in each of these sequences. (i) 8, 14, 20, 26, … … , … [2] (ii) 12, 10, 7, 3, … … , … [2] (b) Find the nth term of this sequence. 14, 25, 36, 47, … … [2] (c) Work out the second term of the sequence whose nth term is 5 3 - 2n ^ h. … [1] (d) 1, 4, 9, 16, … The nth term of this sequence is n2. Use this information to write down the nth term of each of these sequences. (i) 2, 5, 10, 17, … … [1] (ii) 3, 12, 27, 48, … … [1]

9 marks

Mark scheme: 9(a)(i) 32 1 38 1FT FT their 32 + 6 9(a)(ii) –2 1 –8 1FT FT their –2 – 6 9(b) 11n + 3 oe final answer 2 B1 for 11n + k (k may be 0) or jn + 3 (j ≠ 0) or 11n + 3 or 14 + 11(n – 1) seen but not as final answer 9(c) –5 1 9(d)(i) n2 + 1 oe 1 9(d)(ii) 3n2 oe 1

This question in 0580/32 May/June 2017

10 (a) These are the first four terms of a sequence. –2 6 14 22 (i) Write down the next term. … [1] (ii) Write down the rule for continuing the sequence. … [1] (iii) Find an expression for the nth term. … [2] (b) The nth term of another sequence is 5 n + 1 - 6 . ^ h Write down the second term of this sequence. … [1] (c) These are the first four terms of a different sequence. –2 1 8 19 Write down the next term of this sequence. … [1]

6 marks

Mark scheme: 10(a)(i) 30 1 10(a)(ii) add 8 oe 1 10(a)(iii) 8n – 10 oe final answer 2 B1 for 8n + j or kn – 10 (k ≠ 0) 10(b) 9 1 10(c) 34 1

This question in 0580/31 Oct/Nov 2017

9 (a) These are the first four terms of a sequence. 8 15 22 29 (i) Find the next term of this sequence. … [1] (ii) Describe the rule for continuing this sequence. … [1] (iii) Find an expression for the nth term of this sequence. … [2] (b) Find the first three terms of another sequence whose nth term is n 2 + 10 . … , … , … [2] (c) Write down an expression for the nth term of this sequence. 1 8 27 64 … [1]

7 marks

Mark scheme: 9(a)(i) 36 1 9(a)(ii) add 7 oe 1 9(a)(iii) 7n + 1 oe final answer 2 B1 for 7n + c or kn + 1 ( k ≠ 0) or 7n + 1 or 8 + (n – 1)7 spoilt 9(b) 11 14 19 2 B1 for 2 correct If 0 scored SC1 for 10, 11, 14 9(c) n3 1

This question in 0580/33 May/June 2018

4 (a) The diagram shows the first three patterns in a sequence. Pattern 1 Pattern 2 Pattern 3 Pattern 4 On the grid, draw pattern 4. [1] (b) These are the first four terms of another sequence. 41 35 29 23 (i) Write down the next two terms. … , … [2] (ii) Write down the rule for continuing this sequence. … [1] (c) These are the first four terms of a different sequence. 11 15 19 23 (i) Write down an expression for the nth term. … [2] (ii) Is 129 a term in this sequence? Show how you decide. … because … [2]

8 marks

Mark scheme: 4(a) Correct pattern drawn 1 4(b)(i) 17 11 2 B1 for 17 in correct position If 0 scored SC1 for their 17 – 6 correct 4(b)(ii) Subtract 6 oe 1 4(c)(i) 4n + 7 oe final answer 2 B1 for 4n + k or jn + 7, j ≠ 0 4(c)(ii) No, [because] 30.5 is not a whole number 2 M1 for their (c)(i) = 129 or better oe

This question in 0580/33 Oct/Nov 2018

5 Mrs Verma has a restaurant. In the restaurant each table has 8 chairs. Sometimes she puts tables together. The diagrams show how the tables are put together and the position of each chair (X). X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X 1 table 2 tables 3 tables 4 tables The pattern of tables and chairs forms a sequence. (a) Draw the diagram for 4 tables. [1] (b) Complete the table. Number of 1 2 3 4 5 6 tables (t) Number of 8 10 12 chairs (c) [2] (c) Find a formula for the number of chairs, c, in terms of the number of tables, t. c = … [2] (d) 18 tables are put together in this way. Work out the number of chairs needed. … [2] (e) Work out the number of tables, put together in this way, when 80 chairs are needed. … [2]

9 marks

Mark scheme: 5(a) 4 tables and 14 chairs correctly drawn 1 5(b) 14, 16, 18 2 B1 for 2 correct or k, k + 2, k + 4 5(c) 2t + 6 oe final answer 2 B1 for 2t + j or kt + 6 , k ≠ 0 5(d) 42 cao 2 M1 for 18 correctly substituted into their (c) , provided a linear expression 5(e) 37 cao 2 M1 for their (c) = 80

This question in 0580/32 Feb/March 2019

5 (a) The diagram shows a rectangle with length 7a and width 2a. 7a NOT TO SCALE 2a Write an expression, in its simplest form, for (i) the perimeter, … [2] (ii) the area. … [2] (b) The nth term of a sequence is n2 + 5. Find the first three terms of this sequence. … , … , … [2] 12(c) (i) Complete the table of values for y = , x ! 0 . x x -6 -4 -3 -2 -1 1 2 3 4 6 y -2 -3 12 2 [3] 12 (ii) On the grid, draw the graph of y = for -6 G x G -1 and 1 G x G 6. x y 12 10 8 6 4 2 – 6 – 5 – 4 – 3 – 2 – 1 0 1 2 3 4 5 6 x – 2 – 4 – 6 – 8 – 10 –12 [4] (iii) On the grid, draw the line y = 8. [1] 12 (iv) Use your graph to solve = 8. x x = … [1]

15 marks

Mark scheme: 5(a)(i) 18a final answer 2 M1 for 2 × (7a + 2a) oe 5(a)(ii) 14a2 final answer 2 M1 for 7a × 2a 5(b) 6 9 14 2 B1 for 2 correct or 5 6 9 5(c)(i) −4 −6 −12 6 4 3 3 B2 for 4 or 5 correct or B1 for 2 or 3 correct 5(c)(ii) Correct curve 4 B3FT for 9 or 10 points correctly plotted or B2FT for 7 or 8 points correctly plotted or B1FT for 5 or 6 points correctly plotted 5(c)(iii) Correct ruled line drawn 1 5(c)(iv) 1.3 to 1.7 1 FT their curve and their line

This question in 0580/31 May/June 2019

9 (a) These are the first four terms of a sequence. 29 32 35 38 (i) Write down the next term. … [1] (ii) Write down the rule for continuing this sequence. … [1] (b) The nth term of another sequence is n 2 + 5 . (i) Find the first three terms. … , … , … [2] (ii) Show that 261 is a term in this sequence. … … [2] (c) These are the first four terms of a different sequence. 27 33 39 45 Find the nth term of this sequence. … [2]

8 marks

Mark scheme: 9(a)(i) 41 1 9(a)(ii) Add 3 oe 1 9(b)(i) 6, 9, 14 2 B1 for one correct term in correct position If 0 scored, SC1 for 5, 6, 9 9(b)(ii) n2 + 5 = 261 or 261 – 5 = 256 or M1 256 + 5 = 261 or 261 − 5 ( n = ) 256 = 16 A1 or 256 is a square number 9(c) 6n + 21 oe final answer 2 M1 for 6n + j or kn + 21 k ≠ 0

This question in 0580/31 Oct/Nov 2019

7 (a) Here are the first four terms of a sequence. 32 27 22 17 (i) Write down the next term. … [1] (ii) Write down the rule for continuing the sequence. … [1] (b) The nth term of another sequence is n 2 + 2 n . Find the first three terms of this sequence. … , … , … [2] (c) Here are the first three patterns in a sequence. Pattern 1 Pattern 2 Pattern 3 (i) Complete the table. Pattern 1 2 3 4 5 Number of lines 6 [2] (ii) Find an expression, in terms of n, for the number of lines in Pattern n. … [2] (iii) Jake says that he can make one of these patterns using exactly 105 lines. Explain, without doing any working, why he is wrong. … [1]

9 marks

Mark scheme: 7(a)(i) 12 1 7(a)(ii) Subtract 5 oe 1 7(b) 3 8 15 2 B1 for two correct in correct positions If 0 scored, SC1 for 0 3 8 7(c)(i) 10 14 18 22 2 B1 for 2 or 3 correct 7(c)(ii) 4n + 2 oe final answer 2 B1 for 4n + j or kn + 2, k ≠ 0 7(c)(iii) All patterns use an even number of 1 lines oe

This question in 0580/32 Oct/Nov 2019

8 The grid shows the first three diagrams in a sequence. Each diagram is made using small squares that are white or grey. Diagram 1 Diagram 2 Diagram 3 Diagram 4 (a) On the grid, draw Diagram 4. [1] (b) Write down the term to term rule for the number of grey squares. … [1] (c) Diagram number 1 2 3 4 n Number of small white squares 1 4 9 Number of small grey squares 3 5 7 Total number of small squares 4 9 16 Complete the table. [6] (d) Work out the number of small white squares in Diagram 18. … [1] (e) One of the diagrams has a total of 900 small squares. Work out its Diagram number. Diagram … [2] (f) Another diagram has 43 small grey squares. Work out the total number of small squares in this diagram. … [3]

14 marks

Mark scheme: 8(a) Correct diagram 1 8(b) Add 2 oe 1 8(c) 16 n2 oe 6 B2 for 16, 9 and 25 9 2n + 1 oe or B1 for 2 correct 25 (n + 1)2 oe B1 for n2 oe B2 for 2n + 1 oe or B1 for 2n + c or kn + 1 (k ≠ 0) B1 for (n + 1)2 oe 8(d) 324 1 8(e) 29 2 M1 for their (n + 1)2 = 900 or B1 for 900 or 30 8(f) 484 3 M1 for their ( 2 n + 1) = 43 M1dep on M1 scored for 43 + their n 2 2 or for their( n + 1)

This question in 0580/32 Feb/March 2020

10 (a) These are the first four terms of a sequence. 8 15 22 29 (i) Write down the next term. … [1] (ii) Write down the term to term rule for continuing this sequence. … [1] (iii) Find an expression for the nth term. … [2] (b) Find the next term in each of these sequences. (i) 18, 21, 26, 33, 42, … … [1] (ii) 18, 20, 24, 32, 48, … … [1] (c) Find the first three terms of the sequence with nth term n 2 + 5n . … , … , … [2]

8 marks

Mark scheme: 10(a)(i) 36 1 10(a)(ii) add 7 oe 1 10(a)(iii) 7n + 1 oe final answer 2 B1 for 7n + j or kn + 1 (k ≠ 0) as final answer, or for 7n + 1 seen then spoilt 10(b)(i) 53 1 10(b)(ii) 80 1 10(c) 6 14 24 2 B1 for 2 correct terms in the correct place If 0 scored, SC1 for 0 6 14

This question in 0580/31 Oct/Nov 2020

9 A sequence of patterns is made using black counters and white counters. Pattern 1 Pattern 2 Pattern 3 Pattern 4 (a) Draw Pattern 4. [1] (b) Complete the table. Pattern 1 2 3 4 5 Number of black counters 4 6 8 Number of white counters 1 4 9 [2] (c) Write an expression, in terms of n, for (i) the number of black counters in Pattern n, … [2] (ii) the number of white counters in Pattern n. … [1] (d) Elena has 30 black counters and 140 white counters. Can she make Pattern 12 using her counters? Explain your answer. … because … … [2]

8 marks

Mark scheme: 9(a) 1 9(b) 10 12 2 B1 for 2 or 3 correct 16 25 9(c)(i) 2n + 2 oe final answer 2 B1 for 2n + c or kn + 2, (k ≠ 0) as final answer or for 2n + 2 seen then spoilt 9(c)(ii) n2 1 9(d) No 2 M1 for 12 substituted into their 2n + 2 or with a correct supporting reason their n2 or 26 [black] or 144 [white] or 140 = 11.8... [white]

This question in 0580/33 Oct/Nov 2020

9 A sequence of patterns is made using rectangular blocks. Pattern 1 Pattern 2 Pattern 3 Pattern 4 (a) Draw Pattern 4. [1] (b) Complete the table. Pattern number 1 2 3 4 5 Number of blocks 1 4 7 [2] (c) Find an expression, in terms of n, for the number of blocks in Pattern n. … [2] (d) Tara wants to make one pattern in this sequence. She has 84 blocks. Work out the largest pattern number she can make and the number of blocks remaining. Pattern number … Number of blocks remaining … [4]

9 marks

Mark scheme: 9(a) Correct pattern 4 1 9(b) 10 13 2 B1 for each If 0 scored, SC1 for Pattern 5 three more than their Pattern 4 9(c) 3n – 2 oe final answer 2 B1 for 3n + j (j ≠ −2) or kn – 2 (k ≠ 0 or 3) or 3n – 2 oe seen then spoilt 9(d) 28 nfww 4 FT 2 B3 for 28 nfww as answer or B3FT for their n correctly truncated or B2 for 28.6 to 28.7 ( 84 − their (b) ) or B2FT for n = correctly their (a) evaluated or M1 for their (c) = 84 or 3 × 28 – 2 = 82 or for adding up in threes up to 82 or 85

This question in 0580/32 May/June 2021

7 (a) Shade some squares so that both shapes have the same fraction shaded. [2] (b) Here is a pattern. ………. ……….. Position number 1 is a . Position number 2 is a . (i) Draw the next two shapes in this pattern. [1] (ii) What do the position numbers of the shape have in common? … [1] (iii) Pierre says that the shape in position number 99 is a . Explain why he is correct. … … [2] (c)  = { , , , , , , , , , , , } This universal set has twelve elements. Each shape is: • a circle, C, or a triangle, T, or a rectangle, R • large, L, or small, S • black, B, or white, W. (i) B C The triangles and rectangles are drawn in the Venn diagram. (a) Draw the four circles to complete the Venn diagram. [1] (b) Find n ( B , C ) . … [1] (ii) Six of the twelve shapes are drawn in another Venn diagram. … … Complete the Venn diagram by: • labelling the sets and • drawing the shapes , , , , and . [3]

11 marks

Mark scheme: 7(a) 15 squares shaded 2 15 B1 for 15 or 36 5 k or M1 for  oe 12 36 7(b)(i) 1 7(b)(ii) Divisible by 4 oe 1 7(b)(iii) Justifies why Pierre is correct 2 B1 for e.g. 100th term is a e.g. 4  25=100 so the 100th term is a the 99th term is the term before since is always before , Pierre is correct. 7(c)(i)(a) 1 7(c)(i)(b) 8 1 FT their (c)(i)(a) 7(c)(ii) 3 B1 for correct labelling L B B2 for 6 shapes placed correctly or B1 for 4 or 5 shapes placed correctly

This question in 0580/32 May/June 2022

10 Meena makes these patterns using dots and lines. Pattern 1 Pattern 2 Pattern 3 Pattern 4 (a) Draw Pattern 4. [1] (b) Complete the table. Pattern 1 2 3 4 Number of dots 3 5 7 Number of lines 5 9 13 [2] (c) (i) Write down the term to term rule for continuing the sequence for the number of dots. … [1] (ii) Find an expression, in terms of n, for the number of dots in Pattern n. … [2] (d) The number of lines in Pattern n is 4n + 1. Meena makes the pattern which has 129 lines. Work out the number of dots she uses to make this pattern. … [4]

10 marks

Mark scheme: 10(a) Pattern 4 drawn correctly 1 10(b) 9 2 B1 for each 17 10(c)(i) Add 2 oe 1 10(c)(ii) 2n + 1 oe final answer 2 B1 for 2n + j or kn + 1 k ≠ 0 as final answer or for 2n + 1 oe seen then spoilt 10(d) 65 nfww 4 M1 for 4n + 1 = 129 A1 for n = 32 nfww M1 for substitution of their 32 into their (c)(ii)

This question in 0580/33 Oct/Nov 2022

9 (a) These are the first four terms of a sequence. 2 8 14 20 (i) Write down the next term. … [1] (ii) Write down the term to term rule for continuing the sequence. … [1] (iii) Find an expression for the nth term. … [2] (b) (i) Find the first three terms of the sequence with nth term n 2 + 5 . … , … , … [2] (ii) These are the first four terms of another sequence. 7 10 15 22 Find an expression for the nth term of this sequence. … [1]

7 marks

Mark scheme: 9(a)(i) 26 1 9(a)(ii) add 6 oe 1 9(a)(iii) 6n – 4 oe final answer 2 B1 for 6n + j or kn – 4 oe (k ≠ 0) as final answer or correct answer seen then spoilt 9(b)(i) 6 9 14 2 B1 for two in the correct place or answer 5 6 9 9(b)(ii) n2 + 6 oe final answer 1

This question in 0580/32 May/June 2023

5 The grid shows the first three diagrams in a sequence. Each diagram is made using sticks. Diagram 1 Diagram 2 Diagram 3 (a) On the grid, draw Diagram 4. Diagram 4 [1] (b) Complete the table. Diagram number 1 2 3 4 5 Number of sticks 5 9 13 [2] (c) (i) Find an expression, in terms of n, for the number of sticks in Diagram n. … [2] (ii) One of the diagrams has 73 sticks. Work out its Diagram number. Diagram … [2] (d) (i) Show that the total number of sticks needed to make the first 3 diagrams is 27. [1] (ii) The total number of sticks needed to make the first k diagrams is 2k 2 + 3k . Show that this expression gives the correct total number of sticks needed to make the first 3 diagrams. [2] (iii) Tobias wants to make the first 10 diagrams. He has already made the first 3 diagrams. He has 240 sticks left to make the remaining 7 diagrams. Work out how many sticks he has left when all 10 diagrams are made. … [4]

14 marks

Mark scheme: 5(a) Correct diagram drawn 1 5(b) 17, 21 2 B1 for each If B0 scored, SC1 for k , k  4 5(c)(i) 4 n  1 oe final answer 2 B1 for 4 n  k or an  1 , a  0 as final answers or for 4 n  1 oe seen and spoilt 5(c)(ii) 18 nfww 2 M1 for 4 n 1 73 or better or M1 for their(c)(i) = 73 5(d)(i) 5  9  13   27  1 5(d)(ii) 2  32 3 3 2 M1 for 2  32 3 3 leading to 27 as final answer 5(d)(iii) 37 4 M3 for 240  2  10 2 3 10  27 oe     or M2 for 2  10 2 3 10  27 oe or M1 for 2  10 2 3 10 oe

This question in 0580/33 May/June 2023

7 These are the first four diagrams in a sequence. The diagrams are made using dots and lines. Diagram 1 Diagram 2 Diagram 3 Diagram 4 (a) Complete the table. Diagram 1 2 3 4 Number of small squares 2 4 6 Number of dots 6 9 12 Number of lines 7 12 17 [2] (b) Complete this statement. A diagram in this sequence cannot have 51 small squares because … … [1] (c) An expression for the number of dots in Diagram n is 3n + 3 . Which diagram has 249 dots? … [2] (d) (i) Find an expression, in terms of n, for the number of lines in Diagram n. … [2] (ii) Find the number of lines in Diagram 41. … [1]

8 marks

Mark scheme: 7(a) 8 2 B1 for 2 correct in correct position in 15 table 22 7(b) 51 is an odd number oe 1 7(c) 82 2 M1 for 3n + 3 = 249 or 3(n + 1) = 249 or better 7(d)(i) 5n + 2 oe final answer 2 B1 for 5n + j or kn + 2 k ≠ 0 as final answer or for 5n + 2 oe seen then spoilt 7(d)(ii) 207 1 FT dep on their(d)(i) in the form an + b with a ≠ 0

This question in 0580/33 Oct/Nov 2023

3 (a) A sequence of shapes is made with squares and triangles. Shape 1 Shape 2 Shape 3 (i) On the grid, draw Shape 3. [1] (ii) Find the number of triangles in Shape 5. … [1] (b) These are the first four terms of a sequence. 32 25 18 11 (i) Find the next two terms. … , … [2] (ii) Write down the term to term rule for this sequence. … [1] (c) (i) 5, 8, 11, 14, … Find the nth term of this sequence. … [2] (ii) 1, 8, 27, 64, … Find the nth term of this sequence. … [1]

8 marks

Mark scheme: 3(a)(i) Correct shape 1 3(a)(ii) 12 1 3(b)(i) 4 −3 2 B1 for each or second number 7 less than first number or for both answers correct but reversed 3(b)(ii) Subtract 7 oe 1 3(c)(i) 3n + 2 oe final answer 2 B1 for answer 3n + c or kn + 2 ( k ≠ 0) or correct answer seen then spoilt 3(c)(ii) n3 oe final answer 1

This question in 0580/32 Oct/Nov 2024

8 These are the first four terms of a sequence. 31 24 17 10 (a) Write down the term-to-term rule for continuing this sequence. … [1] (b) Find the next two terms in this sequence. … , … [2] (c) Find the nth term. … [2]

5 marks

Mark scheme: 8(a) subtract 7 oe 1 8(b) 3 −4 2 B1 for each If 0 scored SC1 for two answers with a difference of −7 8(c) 38 −n7 oe final answer 2 B1 for j − 7 n or 38−kn , k  0 or 38 −n7 seen then spoilt

This question in 0580/32 Feb/March 2025

13 (a) 2, e, f, 20, … A linear sequence has a first term of 2 and a fourth term of 20. The term-to-term rule for this sequence is add k. Work out the values of e, f and k. e = … f = … k = … [3] (b) These are the first four terms of another sequence. 9 7 5 3 Find the nth term. … [2]

5 marks

Mark scheme: 13(a) 8 14 6 3 B1 for each correctly assigned 20 − 2 or M1 for oe 3 13(b) 11 −n2 oe final answer 2 B1 for answer j −n2 or 11−kn , k  0 or 11 −n2 seen then spoilt

This question in 0580/32 May/June 2025

18 (a) The nth term of a sequence is n 2 - 4 . Find the first 3 terms of this sequence. … , … , … [2] (b) These are the first four terms of a different sequence. 5 2 -1 -4 Find the nth term. … [2]

4 marks

Mark scheme: 18(a) –3 0 5 2 B1 for 2 in correct position in final answer 18(b) –3n + 8 oe final answer 2 B1 for final answer of –3n + j or kn + 8 where k ≠ 0 , or –3n + 8 seen then spoilt

This question in 0580/31 Oct/Nov 2025