(a)Write down the next two terms of the sequence.(2)
(b)Describe, in words, the rule for getting from one term to the next.(1)
(Total for Question 1 is 3 marks)
2
A sequence begins 95, 85, 75, 65, ...
(a)Write down the next two terms of the sequence.(2)
(b)Find the 10th term of the sequence.(2)
(Total for Question 2 is 4 marks)
3
Sofia makes patterns from matchsticks. Pattern 1 is a single square. Pattern 2 is two squares in a row, sharing one side with the first square. Pattern 3 is three squares in a row, again sharing sides. The table below shows the number of matchsticks used. Pattern number: 1, 2, 3, 4, 5 Matchsticks: 4, 7, 10, ?, ?
(a)Complete the table for pattern 4 and pattern 5.(2)
(b)Describe, in words, the rule connecting the pattern number and the number of matchsticks.(1)
(Total for Question 3 is 3 marks)
4
A sequence begins 2, 5, 8, __, 14, __. Find the two missing terms.
(Total for Question 4 is 2 marks)
5
Which number continues the sequence? 7, 14, 21, __, 35
A) 26
B) 27
C) 28
D) 29
(Total for Question 5 is 1 mark)
6
A sequence has nth term 4n + 3.
(a)Work out the first three terms of the sequence.(3)
(b)Work out the 20th term of the sequence.(2)
(c)Is 100 a term in this sequence? Give a reason for your answer.(2)
(Total for Question 6 is 7 marks)
7
A sequence begins 5, 9, 13, 17, ...
(a)Write down the term-to-term rule for this sequence.(1)
(b)Find the 8th term of the sequence.(2)
(c)Priya says the 50th term of the sequence is 200. Show that Priya is incorrect, and state the correct value of the 50th term.(2)
(Total for Question 7 is 5 marks)
8
A bakery bakes 3 loaves on day 1. Each day after that, the bakery doubles the number of loaves it baked the day before.
(a)Complete the sequence for days 1 to 5.(2)
(b)Work out how many loaves the bakery will bake on day 8.(2)
(Total for Question 8 is 4 marks)
9
A pattern of dots is arranged so that pattern 1 has 5 dots, pattern 2 has 8 dots and pattern 3 has 11 dots.
(a)Find the number of dots in pattern 6.(2)
(b)Find the pattern number that has exactly 47 dots.(2)
(Total for Question 9 is 4 marks)
10
A sequence begins 2, 5, 7, 12, 19, ... where, from the third term onwards, each term is the sum of the two terms before it.
(a)Write down the next two terms of the sequence.(2)
(b)Describe, in words, the rule connecting the terms of the sequence.(1)
(Total for Question 10 is 3 marks)
11
In a theatre, row 1 has 8 seats. Each row after that has 3 more seats than the row in front of it.
(a)Write down the number of seats in row 2, row 3 and row 4.(2)
(b)Find a formula for the number of seats in row n.(2)
(c)The theatre has 20 rows. Find the number of seats in row 20.(2)
(Total for Question 11 is 6 marks)
12
A sequence begins 20, 15, 10, 5, ...
(a)Write down the next three terms of the sequence.(2)
(b)Write down the term-to-term rule for this sequence.(1)
(c)Find the position of the first term in the sequence that is negative.(2)
(Total for Question 12 is 5 marks)
13
A sequence begins 7, 12, 17, 22, ...
(a)Find the nth term of the sequence.(2)
(b)Use your formula to find the 25th term of the sequence.(2)
(c)Determine whether 152 is a term in the sequence. Show your working.(2)
(Total for Question 13 is 6 marks)
14
Which number continues the sequence? 1, 1, 2, 3, 5, 8, 13, __
A) 18
B) 20
C) 21
D) 24
(Total for Question 14 is 1 mark)
15
Sequence A begins 2, 5, 8, 11, ... with nth term 3n - 1. Sequence B begins 4, 7, 10, 13, ... with nth term 3m + 1. Show that no term of sequence A is ever equal to a term of sequence B.
(Total for Question 15 is 3 marks)
16
A pattern of square grids of dots is arranged so that pattern 1 has 1 dot, pattern 2 has 4 dots, pattern 3 has 9 dots, pattern 4 has 16 dots and pattern 5 has 25 dots.
(a)Write down the next two terms of the sequence.(2)
(b)Explain how the pattern of differences between the terms helps you find the next term.(1)
(c)Find the 12th term of the sequence without listing every term.(2)
(Total for Question 16 is 5 marks)
17
A pattern of dots is arranged in rectangles so that pattern n has n2 + n dots in total. Pattern 1 has 2 dots, pattern 2 has 6 dots, pattern 3 has 12 dots and pattern 4 has 20 dots.
(a)Show that the rule n2 + n gives 12 dots when n = 3.(1)
(b)Find the pattern number that has exactly 90 dots.(3)
(Total for Question 17 is 4 marks)
18
Ffion saves 2 pounds sterling in week 1. Each week after that, she saves 3 pounds sterling more than the week before, so she saves 2, 5, 8, 11, ... pounds sterling.
(a)Find how much Ffion saves in week 6.(2)
(b)Find the week in which Ffion first saves exactly 29 pounds sterling.(2)
(c)Find the total amount Ffion has saved by the end of week 4.(2)
(Total for Question 18 is 6 marks)
19
A sequence is formed using the rule 'double the previous term, then subtract 1', starting at 4. This gives 4, 7, 13, 25, ...
(a)Find the next two terms of the sequence.(2)
(b)A second sequence uses the same rule but starts at 1. Find the first four terms of this sequence, and state what you notice.(2)
(Total for Question 19 is 4 marks)
20
A sequence begins 8, 10, 12, 14, ... Aled says the nth term of this sequence is 2n + 5. Explain why Aled's formula is incorrect, and write down the correct nth term for the sequence.
(Total for Question 20 is 3 marks)
21
A row of hexagons is made from matchsticks. A single hexagon uses 6 matchsticks. Each extra hexagon added to the row shares one side with the hexagon before it, so it only needs 5 more matchsticks. The number of matchsticks in a row of n hexagons is 5n + 1.
(a)Find the number of matchsticks needed for a row of 7 hexagons.(2)
(b)A row of hexagons is made using exactly 151 matchsticks. Find the number of hexagons in the row.(2)
(Total for Question 21 is 4 marks)
Mark scheme · EP.M13 Sequences and Patterns
Question 1
(a) B1 19 oe
(a) B1 23 oe
(a) Answer: 19, 23
(b) B1 add 4 to the previous term (oe)
(b) Answer: Add 4 to the previous term
Question 2
(a) B1 55 oe
(a) B1 45 oe
(a) Answer: 55, 45
(b) M1 valid method, e.g. nth term = 105 - 10n, or systematic continuation to the 10th term
(b) A1 5 cao
(b) Answer: 5
Question 3
(a) B1 13
(a) B1 16
(a) Answer: 13, 16
(b) B1 oe, e.g. start at 4 and add 3 for each extra square, or multiply the pattern number by 3 and add 1
(b) Answer: Start at 4 and add 3 for each extra square (number of matchsticks = 3 x pattern number + 1)
Question 4
B1 11
B1 17
Answer: 11, 17
Question 5
B1 C) 28
Answer: C) 28
Question 6
(a) M1 correct substitution shown for at least one value of n
(a) A1 two of the three terms correct
(a) A1 all three terms correct cao
(a) Answer: 7, 11, 15
(b) M1 4 x 20 + 3 seen
(b) A1 83 cao
(b) Answer: 83
(c) M1 sets 4n + 3 = 100 and attempts to solve for n
(c) A1 correct conclusion with valid reason: No, because n = 24.25, which is not a whole number
(c) Answer: No, because n = 24.25 which is not a whole number
Question 7
(a) B1 add 4 (oe)
(a) Answer: Add 4 to the previous term
(b) M1 valid method, e.g. nth term = 4n + 1, or continued pattern to the 8th term
(b) A1 33 cao
(b) Answer: 33
(c) M1 4 x 50 + 1 seen
(c) A1 cso, 201 stated with correct conclusion that Priya is wrong
(c) Answer: Priya is incorrect; the 50th term is 201
Question 8
(a) B1 first three terms correct: 3, 6, 12
(a) B1 all five terms correct: 3, 6, 12, 24, 48
(a) Answer: 3, 6, 12, 24, 48
(b) M1 valid method, e.g. 3 x 27, or continued doubling to day 8
(b) A1 384 cao
(b) Answer: 384
Question 9
(a) M1 nth term 3n + 2 identified, or valid continuation to pattern 6
(a) A1 20 cao
(a) Answer: 20
(b) M1 sets 3n + 2 = 47 and attempts to solve
(b) A1 15 cao
(b) Answer: Pattern 15
Question 10
(a) B1 31 oe
(a) B1 50 (ft, sum of previous two terms)
(a) Answer: 31, 50
(b) B1 oe: each term (from the third) is the sum of the two terms before it
(b) Answer: Each term after the first two is found by adding together the two terms before it
Question 11
(a) B1 two of the three values correct
(a) B1 all three correct: 11, 14, 17
(a) Answer: 11, 14, 17
(b) M1 valid method using common difference 3 and row 1 value 8
(b) A1 3n + 5 cao
(b) Answer: Number of seats = 3n + 5
(c) M1 their formula with n = 20 substituted (ft)
(c) A1 65 cao
(c) Answer: 65 seats
Question 12
(a) M1 valid method continuing the pattern
(a) A1 all three correct: 0, -5, -10
(a) Answer: 0, -5, -10
(b) B1 subtract 5 (oe)
(b) Answer: Subtract 5 from the previous term
(c) M1 valid method, e.g. nth term = 25 - 5n and solve 25 - 5n < 0, or systematic listing
(c) A1 cao, the 6th term (value -5)
(c) Answer: The 6th term (which is -5)
Question 13
(a) M1 common difference of 5 identified and used
(a) A1 5n + 2 cao
(a) Answer: 5n + 2
(b) M1 their formula with n = 25 substituted (ft)
(b) A1 127 cao
(b) Answer: 127
(c) M1 sets 5n + 2 = 152 and attempts to solve
(c) A1 cso, yes, n = 30, so 152 is the 30th term
(c) Answer: Yes, 152 is the 30th term
Question 14
B1 C) 21
Answer: C) 21
Question 15
M1 sets 3n - 1 = 3m + 1
M1 rearranges to 3n - 3m = 2, or 3(n - m) = 2 (oe)
A1 cso: the left-hand side is always a multiple of 3, but 2 is not a multiple of 3, so there is no integer solution and the sequences never share a term
Answer: No common term exists, since 3(n - m) = 2 has no integer solution
Question 16
(a) B1 36
(a) B1 49
(a) Answer: 36, 49
(b) B1 oe: the differences are consecutive odd numbers (3, 5, 7, 9, 11, ...), each 2 more than the last, because each term is a square number
(b) Answer: The differences are consecutive odd numbers (3, 5, 7, 9, 11, ...), each 2 more than the last, because each term is a square number
(c) M1 identifies nth term = n2
(c) A1 144 cao
(c) Answer: 144
Question 17
(a) B1 cso: 32 + 3 = 9 + 3 = 12
(a) Answer: 32 + 3 = 9 + 3 = 12
(b) M1 sets n2 + n = 90, i.e. n2 + n - 90 = 0
(b) M1 solves by factorising or trial, e.g. (n - 9)(n + 10) = 0
(b) A1 cao, n = 9 (rejecting n = -10)
(b) Answer: Pattern 9
Question 18
(a) M1 nth term 3n - 1 identified, or valid continuation to week 6
(a) A1 17 pounds sterling cao
(a) Answer: 17 pounds sterling
(b) M1 sets 3n - 1 = 29 and attempts to solve
(b) A1 week 10 cao
(b) Answer: Week 10
(c) M1 adds the first four terms: 2 + 5 + 8 + 11
(c) A1 26 pounds sterling cao
(c) Answer: 26 pounds sterling
Question 19
(a) M1 applies the rule correctly at least once (ft)
(a) A1 49 and 97 cao
(a) Answer: 49, 97
(b) B1 first four terms correctly given as 1, 1, 1, 1
(b) B1 correct observation, e.g. the sequence stays the same (constant)
(b) Answer: 1, 1, 1, 1; the sequence stays the same (constant) because 2 x 1 - 1 = 1
Question 20
M1 tests Aled's formula, e.g. substitutes n = 1 into 2n + 5 to get 7, and compares with the actual first term 8, showing a mismatch
M1 finds the correct constant using the common difference of 2 and first term 8
A1 correct formula 2n + 6 cao
Answer: Aled's formula is incorrect because 2(1) + 5 = 7, not 8; the correct nth term is 2n + 6