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13+ · Mathematics

CE.M9 Sequences, Patterns and Formulae

ISEB (INDEPENDENT SCHOOLS EXAMINATIONS BOARD) - COMMON ENTRANCE ISEB CE 13+ Mathematics · Calculator allowed · about 70 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

Here is a number sequence:
3, 7, 11, 15, ...
Write down the next two terms of the sequence.

(Total for Question 1 is 2 marks)

A sequence begins:
2, 6, 18, 54, ...

(a)Write down the term-to-term rule for this sequence.(1)

(b)Find the next term in the sequence.(1)

(Total for Question 2 is 2 marks)

A pattern is made from rows of triangles using matchsticks:
1 triangle uses 3 matchsticks.
A row of 2 triangles uses 5 matchsticks.
A row of 3 triangles uses 7 matchsticks.
How many matchsticks are needed for a row of 4 triangles?

(Total for Question 3 is 2 marks)

Fill in the two missing terms in this sequence:
5, __, 17, __, 29

(Total for Question 4 is 3 marks)

Which term-to-term rule describes this sequence?
100, 90, 80, 70, ...

A) Add 10
B) Subtract 10
C) Multiply by 10
D) Divide by 10

(Total for Question 5 is 1 mark)

Find a formula for the nth term of this sequence:
4, 9, 14, 19, ...

(Total for Question 6 is 3 marks)

The nth term of a sequence is 3n + 2. Find the 15th term of the sequence.

(Total for Question 7 is 2 marks)

The nth term of a sequence is 4n - 3.
Determine whether 100 is a term of this sequence. Show your working.

(Total for Question 8 is 3 marks)

Matchstick squares are joined in a row.
A row of 1 square uses 4 matchsticks.
A row of 2 squares uses 7 matchsticks.
A row of 3 squares uses 10 matchsticks.

(a)How many matchsticks are needed for a row of 4 squares?(2)

(b)Find a formula for the number of matchsticks, m, needed for a row of n squares.(2)

(Total for Question 9 is 4 marks)

A plumber charges using the formula C = 35 + 40h, where C is the total cost in pounds and h is the number of hours worked.
Calculate the cost of a job that takes 3.5 hours.

(Total for Question 10 is 3 marks)

Make a the subject of the formula:
v = u + at

(Total for Question 11 is 3 marks)

A gardener charges a call-out fee of £15 plus £12 for every hour of work.

(a)Write a formula for the total cost, C pounds, for h hours of work.(2)

(b)Bill received a bill for £87. Calculate how many hours the gardener worked.(2)

(Total for Question 12 is 4 marks)

Triangular numbers are formed by the rule n(n+1)/2.

(a)Write down the first four triangular numbers.(1)

(b)Calculate the 6th triangular number.(2)

(Total for Question 13 is 3 marks)

Here is a sequence:
3, 8, 15, 24, 35, ...
Find a formula for the nth term of this sequence.

(Total for Question 14 is 5 marks)

The nth term of a sequence is n² - n + 1.

(a)Find the 1st term and the 5th term of the sequence.(2)

(b)A student says, 'I think 150 is a term in this sequence.' Show that the student is wrong.(2)

(Total for Question 15 is 4 marks)

A sequence begins:
5, 15, 45, 135, ...

(a)State the term-to-term rule for this sequence.(1)

(b)Find the 7th term of the sequence.(3)

(Total for Question 16 is 4 marks)

The area of a trapezium is given by A = (1/2)(a+b)h, where a and b are the lengths of the parallel sides and h is the height.

(a)Make h the subject of the formula.(3)

(b)A trapezium has area 60 cm² and parallel sides of length 8 cm and 12 cm. Calculate its height.(2)

(Total for Question 17 is 5 marks)

Aisha saves money each month. In month 1 she saves £20. Each month after that, she saves £15 more than the month before.

(a)Find a formula for the amount, A pounds, Aisha saves in month n.(2)

(b)Calculate the total amount Aisha has saved by the end of month 6.(3)

(Total for Question 18 is 5 marks)

The nth term of a sequence is 2n² - 1.

(a)Find the first four terms of the sequence.(2)

(b)Find the differences between consecutive terms, and show that the second difference is constant.(2)

(Total for Question 19 is 4 marks)

A theatre has rows of seats. Row 1 has 8 seats, row 2 has 11 seats, row 3 has 14 seats, and so on, with each row having 3 more seats than the row before.

(a)Find a formula for the number of seats in row n.(2)

(b)The theatre has 20 rows. Calculate the number of seats in row 20.(2)

(c)The manager says, 'No row has exactly 100 seats.' Determine whether the manager is correct.(2)

(Total for Question 20 is 6 marks)

Mark scheme · CE.M9 Sequences, Patterns and Formulae

Question 1

B1 19
B1 23
Answer: 19, 23

Question 2

(a) B1 multiply by 3 (oe: x3)
(a) Answer: Multiply by 3
(b) B1 162 (ft from part a)
(b) Answer: 162

Question 3

M1 identifies the pattern increases by 2 matchsticks each time
A1 9 (cao)
Answer: 9

Question 4

M1 finds common difference = 6, e.g. (17-5)/2 = 6
A1 11
A1 23
Answer: 11, 23

Question 5

B1 B
Answer: B

Question 6

M1 common difference = 5
M1 sets up nth term in the form 5n + c
A1 nth term = 5n - 1 (oe)
Answer: 5n - 1

Question 7

M1 substitutes n = 15 into 3n + 2
A1 47 (cao)
Answer: 47

Question 8

M1 sets 4n - 3 = 100
M1 solves to n = 25.75
A1 concludes 100 is not a term, since n is not a whole number
Answer: No, 100 is not a term (n = 25.75, which is not a whole number)

Question 9

(a) M1 continues the pattern by adding 3 matchsticks
(a) A1 13 (cao)
(a) Answer: 13
(b) M1 common difference = 3
(b) A1 m = 3n + 1 (oe)
(b) Answer: m = 3n + 1

Question 10

M1 substitutes h = 3.5 into C = 35 + 40h
A1 40 x 3.5 = 140 (oe)
A1 C = 175 (cao)
Answer: £175

Question 11

M1 subtracts u from both sides: v - u = at
M1 divides both sides by t
A1 a = (v-u)/t (oe)
Answer: a = (v-u)/t

Question 12

(a) M1 identifies the fixed fee of 15 and rate of 12 per hour
(a) A1 C = 15 + 12h (oe)
(a) Answer: C = 15 + 12h
(b) M1 87 - 15 = 72 (or sets up equation 87 = 15 + 12h)
(b) A1 h = 6 (cao)
(b) Answer: 6 hours

Question 13

(a) B1 1, 3, 6, 10 (all four correct)
(a) Answer: 1, 3, 6, 10
(b) M1 substitutes n = 6 into n(n+1)/2
(b) A1 21 (cao)
(b) Answer: 21

Question 14

M1 finds first differences 5, 7, 9, 11
M1 finds second difference = 2, so the n² coefficient is 1
M1 subtracts n² from the sequence to find the linear part: 2, 4, 6, 8, 10 (= 2n)
A1 nth term = n² + 2n (oe)
B1 verifies with a term, e.g. n=5: 25+10=35
Answer: n² + 2n

Question 15

(a) B1 1st term = 1
(a) B1 5th term = 21
(a) Answer: 1st term = 1, 5th term = 21
(b) M1 finds the 12th term (133) and the 13th term (157)
(b) A1 cso: concludes 150 lies strictly between two consecutive terms, so it cannot be a term
(b) Answer: 150 is not a term: the 12th term is 133 and the 13th term is 157, and 150 falls between them

Question 16

(a) B1 multiply by 3 (oe: x3)
(a) Answer: Multiply by 3
(b) M1 continues the sequence correctly
(b) A1 ft: 6th term = 1215
(b) A1 ft: 7th term = 3645
(b) Answer: 3645

Question 17

(a) M1 multiplies both sides by 2: 2A = (a+b)h
(a) M1 divides both sides by (a+b)
(a) A1 h = 2A/(a+b) (oe)
(a) Answer: h = 2A/(a+b)
(b) M1 substitutes A=60, a=8, b=12 into h=2A/(a+b)
(b) A1 h = 6 cm (cao)
(b) Answer: 6 cm

Question 18

(a) M1 common difference = 15
(a) A1 A = 15n + 5 (oe)
(a) Answer: A = 15n + 5
(b) M1 lists the six monthly amounts, or uses sum = (n/2)(2a+(n-1)d)
(b) A1 correct process, e.g. 3(40+75) = 345
(b) A1 total = £345 (cao)
(b) Answer: £345

Question 19

(a) M1 substitutes n = 1, 2, 3, 4
(a) A1 1, 7, 17, 31 (all correct)
(a) Answer: 1, 7, 17, 31
(b) M1 first differences 6, 10, 14
(b) A1 second differences both = 4, so the second difference is constant
(b) Answer: First differences: 6, 10, 14. Second difference: 4 (constant)

Question 20

(a) M1 common difference = 3
(a) A1 seats = 3n + 5 (oe)
(a) Answer: 3n + 5
(b) M1 substitutes n = 20 into 3n + 5
(b) A1 65 (cao)
(b) Answer: 65
(c) M1 sets 3n + 5 = 100 and solves to n = 31.67 (or 31 and 2/3, oe)
(c) A1 concludes the manager is correct, since n is not a whole number
(c) Answer: The manager is correct: n = 31.67 (recurring), which is not a whole number

Sequences, Patterns and Formulae - Worksheets, Questions and Revision