(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 120, 112, 104, 96, ...
(a)Write down the next two terms of the sequence.(2)
(b)Find the 12th term of the sequence.(2)
(Total for Question 2 is 4 marks)
3
A cafe pushes small square tables together in a row to seat a group. One table on its own seats 4 people. Two tables pushed together in a row seat 6 people. Three tables pushed together in a row seat 8 people. The table below shows the number of chairs used. Number of tables: 1, 2, 3, 4, 5 Chairs: 4, 6, 8, ?, ?
(a)Complete the table for 4 tables and 5 tables.(2)
(b)Describe, in words, the rule connecting the number of tables and the number of chairs.(1)
(Total for Question 3 is 3 marks)
4
A sequence begins 9, 15, 21, __, 33, __. Find the two missing terms.
(Total for Question 4 is 2 marks)
5
Which number continues the sequence? 9, 18, 27, __, 45
A) 34
B) 35
C) 36
D) 37
(Total for Question 5 is 1 mark)
6
Which number continues the sequence? 100, 91, 82, __, 64
A) 70
B) 71
C) 72
D) 73
(Total for Question 6 is 1 mark)
7
A bike hire scheme charges a fixed booking fee plus an hourly rate. The cost, in pounds sterling, of hiring a bike for n hours is given by 6n + 2.
(a)Work out the cost of hiring a bike for 1 hour, 2 hours and 3 hours.(3)
(b)Work out the cost of hiring a bike for 25 hours.(2)
(c)Is it possible to hire a bike for a whole number of hours for exactly 250 pounds sterling? Give a reason for your answer.(2)
(Total for Question 7 is 7 marks)
8
A sequence begins 8, 13, 18, 23, ...
(a)Write down the term-to-term rule for this sequence.(1)
(b)Find the 9th term of the sequence.(2)
(c)Kwame says the 40th term of the sequence is 200. Show that Kwame is incorrect, and state the correct value of the 40th term.(2)
(Total for Question 8 is 5 marks)
9
A video posted online gets 5 views on day 1. Each day after that, the number of views doubles.
(a)Complete the sequence for days 1 to 5.(2)
(b)Work out how many views the video will have on day 7.(2)
(Total for Question 9 is 4 marks)
10
A pattern of tiles is arranged in a plus (cross) shape. Pattern 1 uses 5 tiles, pattern 2 uses 9 tiles and pattern 3 uses 13 tiles.
(a)Find the number of tiles in pattern 7.(2)
(b)Find the pattern number that has exactly 57 tiles.(2)
(Total for Question 10 is 4 marks)
11
A sequence begins 3, 4, 7, 11, 18, ... 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 11 is 3 marks)
12
A multi-storey car park has 12 spaces on level 1. Each level above that has 7 more spaces than the level below it.
(a)Write down the number of spaces on level 2, level 3 and level 4.(2)
(b)Find a formula for the number of spaces on level n.(2)
(c)The car park has 15 levels. Find the number of spaces on level 15.(2)
(Total for Question 12 is 6 marks)
13
A sequence begins 30, 24, 18, 12, ...
(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 13 is 5 marks)
14
A sequence begins 9, 16, 23, 30, ...
(a)Find the nth term of the sequence.(2)
(b)Use your formula to find the 22nd term of the sequence.(2)
(c)Determine whether 191 is a term in the sequence. Show your working.(2)
(Total for Question 14 is 6 marks)
15
Which number continues the sequence? 2, 2, 4, 6, 10, 16, 26, __
A) 36
B) 40
C) 42
D) 44
(Total for Question 15 is 1 mark)
16
Sequence C begins 3, 7, 11, 15, ... with nth term 4n - 1. Sequence D begins 6, 10, 14, 18, ... with nth term 4m + 2. Show that no term of sequence C is ever equal to a term of sequence D.
(Total for Question 16 is 3 marks)
17
Oranges are stacked in a triangular display. Pattern 1 has 1 orange, pattern 2 has 3 oranges and pattern 3 has 6 oranges, each new pattern adding one more row than the last.
(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 number of oranges in pattern 12 without listing every term.(2)
(Total for Question 17 is 5 marks)
18
A garden centre lays rectangular patios from square slabs. Patio n is n slabs wide and (n + 2) slabs long, so the total number of slabs used is n2 + 2n. Patio 1 uses 3 slabs, patio 2 uses 8 slabs and patio 3 uses 15 slabs.
(a)Show that the rule n2 + 2n gives 15 slabs when n = 3.(1)
(b)Find the patio number that uses exactly 99 slabs.(3)
(Total for Question 18 is 4 marks)
19
Owen collects football stickers. He collects 3 stickers in week 1. Each week after that, he collects 4 more stickers than the week before, so he collects 3, 7, 11, 15, ... stickers.
(a)Find how many stickers Owen collects in week 7.(2)
(b)Find the week in which Owen first collects exactly 39 stickers.(2)
(c)Find the total number of stickers Owen has collected by the end of week 5.(2)
(Total for Question 19 is 6 marks)
20
A sequence is formed using the rule 'double the previous term, then add 3', starting at 2. This gives 2, 7, 17, 37, ...
(a)Find the next two terms of the sequence.(2)
(b)A second sequence uses the same rule but starts at -3. Find the first four terms of this sequence, and state what you notice.(2)
(Total for Question 20 is 4 marks)
21
A sequence begins 11, 15, 19, 23, ... Amina says the nth term of this sequence is 4n + 8. Explain why Amina's formula is incorrect, and write down the correct nth term for the sequence.
(Total for Question 21 is 3 marks)
22
A row of kite-shaped tiles is laid along a garden path. A single kite tile has 4 edges. Each extra kite tile added to the row shares one edge with the tile before it, so it only needs 3 more edges. The number of edges in a row of n kite tiles is 3n + 1.
(a)Find the number of edges needed for a row of 9 kite tiles.(2)
(b)A row of kite tiles is made using exactly 88 edges. Find the number of kite tiles in the row.(2)
(Total for Question 22 is 4 marks)
Mark scheme · EP.M13B Sequences and Patterns: Practice Set 2
Question 1
(a) B1 29 oe
(a) B1 35 oe
(a) Answer: 29, 35
(b) B1 add 6 to the previous term (oe)
(b) Answer: Add 6 to the previous term
Question 2
(a) B1 88 oe
(a) B1 80 oe
(a) Answer: 88, 80
(b) M1 valid method, e.g. nth term = 128 - 8n, or systematic continuation to the 12th term
(b) A1 32 cao
(b) Answer: 32
Question 3
(a) B1 10
(a) B1 12
(a) Answer: 10, 12
(b) B1 oe, e.g. start at 4 and add 2 for each extra table, or double the number of tables and add 2
(b) Answer: Start at 4 and add 2 for each extra table (number of chairs = 2 x number of tables + 2)
Question 4
B1 27
B1 39
Answer: 27, 39
Question 5
B1 C) 36
Answer: C) 36
Question 6
B1 D) 73
Answer: D) 73
Question 7
(a) M1 correct substitution shown for at least one value of n
(c) M1 sets 6n + 2 = 250 and attempts to solve for n
(c) A1 correct conclusion with valid reason: No, because n = 41.33..., which is not a whole number
(c) Answer: No, because n = 41.33... which is not a whole number
Question 8
(a) B1 add 5 (oe)
(a) Answer: Add 5 to the previous term
(b) M1 valid method, e.g. nth term = 5n + 3, or continued pattern to the 9th term
(b) A1 48 cao
(b) Answer: 48
(c) M1 5 x 40 + 3 seen
(c) A1 cso, 203 stated with correct conclusion that Kwame is wrong
(c) Answer: Kwame is incorrect; the 40th term is 203
Question 9
(a) B1 first three terms correct: 5, 10, 20
(a) B1 all five terms correct: 5, 10, 20, 40, 80
(a) Answer: 5, 10, 20, 40, 80
(b) M1 valid method, e.g. 5 x 26, or continued doubling to day 7
(b) A1 320 cao
(b) Answer: 320
Question 10
(a) M1 nth term 4n + 1 identified, or valid continuation to pattern 7
(a) A1 29 cao
(a) Answer: 29
(b) M1 sets 4n + 1 = 57 and attempts to solve
(b) A1 14 cao
(b) Answer: Pattern 14
Question 11
(a) B1 29 oe
(a) B1 47 (ft, sum of previous two terms)
(a) Answer: 29, 47
(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 12
(a) B1 two of the three values correct
(a) B1 all three correct: 19, 26, 33
(a) Answer: 19, 26, 33
(b) M1 valid method using common difference 7 and level 1 value 12
(b) A1 7n + 5 cao
(b) Answer: Number of spaces = 7n + 5
(c) M1 their formula with n = 15 substituted (ft)
(c) A1 110 cao
(c) Answer: 110 spaces
Question 13
(a) M1 valid method continuing the pattern
(a) A1 all three correct: 6, 0, -6
(a) Answer: 6, 0, -6
(b) B1 subtract 6 (oe)
(b) Answer: Subtract 6 from the previous term
(c) M1 valid method, e.g. nth term = 36 - 6n and solve 36 - 6n < 0, or systematic listing
(c) A1 cao, the 7th term (value -6)
(c) Answer: The 7th term (which is -6)
Question 14
(a) M1 common difference of 7 identified and used
(a) A1 7n + 2 cao
(a) Answer: 7n + 2
(b) M1 their formula with n = 22 substituted (ft)
(b) A1 156 cao
(b) Answer: 156
(c) M1 sets 7n + 2 = 191 and attempts to solve
(c) A1 cso, yes, n = 27, so 191 is the 27th term
(c) Answer: Yes, 191 is the 27th term
Question 15
B1 C) 42
Answer: C) 42
Question 16
M1 sets 4n - 1 = 4m + 2
M1 rearranges to 4n - 4m = 3, or 4(n - m) = 3 (oe)
A1 cso: the left-hand side is always a multiple of 4, but 3 is not a multiple of 4, so there is no integer solution and the sequences never share a term
Answer: No common term exists, since 4(n - m) = 3 has no integer solution
Question 17
(a) B1 10
(a) B1 15
(a) Answer: 10, 15
(b) B1 oe: the differences are consecutive whole numbers (2, 3, 4, 5, ...), each one more than the last, because each new row has one more orange than the row before it
(b) Answer: The differences are consecutive whole numbers (2, 3, 4, 5, ...), each one more than the last, because each new row has one more orange than the row before it
(c) M1 valid method, e.g. pairing 1 + 2 + ... + 12 as (1 + 12) x 12 / 2, or using pattern n = n(n + 1) / 2
(c) A1 78 cao
(c) Answer: 78
Question 18
(a) B1 cso: 32 + 2 x 3 = 9 + 6 = 15
(a) Answer: 32 + 2 x 3 = 9 + 6 = 15
(b) M1 sets n2 + 2n = 99, i.e. n2 + 2n - 99 = 0
(b) M1 solves by factorising or trial, e.g. (n - 9)(n + 11) = 0
(b) A1 cao, n = 9 (rejecting n = -11)
(b) Answer: Patio 9
Question 19
(a) M1 nth term 4n - 1 identified, or valid continuation to week 7
(a) A1 27 cao
(a) Answer: 27 stickers
(b) M1 sets 4n - 1 = 39 and attempts to solve
(b) A1 week 10 cao
(b) Answer: Week 10
(c) M1 adds the first five terms: 3 + 7 + 11 + 15 + 19
(c) A1 55 cao
(c) Answer: 55 stickers
Question 20
(a) M1 applies the rule correctly at least once (ft)
(a) A1 77 and 157 cao
(a) Answer: 77, 157
(b) B1 first four terms correctly given as -3, -3, -3, -3
(b) B1 correct observation, e.g. the sequence stays the same (constant)
(b) Answer: -3, -3, -3, -3; the sequence stays the same (constant) because 2 x (-3) + 3 = -3
Question 21
M1 tests Amina's formula, e.g. substitutes n = 1 into 4n + 8 to get 12, and compares with the actual first term 11, showing a mismatch
M1 finds the correct constant using the common difference of 4 and first term 11
A1 correct formula 4n + 7 cao
Answer: Amina's formula is incorrect because 4(1) + 8 = 12, not 11; the correct nth term is 4n + 7