O LEVEL (P1)

NUMBER SEQUENCES
QUESTION'S
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1 ( a ) Write down the next two terms in the sequence 20, 16W , 13, 9W , 6, …….
(b) Write down an expression, in terms of n , for the nth term of the sequence

1, 4, 7, 10, 13, ………… .

Answer (a) .................... , .................... [1]

(b) ........................................... [1]

2 It is given that N = 87 × 132.

(a) Complete the statements in the answer space.

Answer (a) 88 × 132 = N + .......................... [1]

87 × 131 = N – .......................... [1]

(b) Hence evaluate 88 × 132 – 87 × 131.

Answer (b) ................................................... [1]

3 (a) The first five terms of a sequence are 1, 3, 6, 10, 15.

1
The nth term of this sequence is n(n + 1).
Find the 19th term. 2

(b) Write down an expression, in terms of n, for the nth term of the sequence

3, 6, 10, 15, 21, ...........

Answer (a) ........................................... [1]

(b) ........................................... [1]

4 A series of diagrams, using three types of triangle,2 is shown below.
The triangles are grey, white or black.

Diagram 1 Diagram 2 Diagram 3 Diagram 4

The table below shows the numbers of each type of triangle used in the diagrams.

Diagram 1 2 3 4 n
Grey triangles 2 4 6 x
White triangles 1 4 9 y
Black triangles 0 2 6 z

(a) Complete the column for Diagram 4. [1]

(b) By considering the number patterns in the table, find, in terms of n, expressions for
x, y and z.

Answer (b) x = ...........................................

y = ...........................................

z = ...........................................[4]
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5 It is given that 68.2 × 0.235 = 16.027.

Hence evaluate

(a) 0.0682 × 2350,

(b) 160.27 ÷ 0.0235.

Answer (a) ............................................[1]

(b) ............................................[1]

6 (a) The nth term of a sequence is 7 – 2n.

Write down the 23rd term in this sequence.

Answer (a) ............................................[1]

(b) (i) The first five terms of another sequence are

4 7 10 13 16.

Write down an expression, in terms of n, for the nth term of this sequence.

(ii) The first five terms of another sequence are

4 7 10 13 16 .
1 4 9 16 25
(a) Write down the next term in this sequence.
(b) Write down an expression, in terms of n, for the nth term of this sequence.

Answer (b) (i) .......................................[1]

(b)(ii)(a) ...................................[1]

(b)(ii)(b) ...................................[1]
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7 The nth term of a sequence is 42 .

n
(a) Write down the first three terms of the sequence, expressing each term in its simplest form.

Answer (a) ........... , ........... , ...........[1]

(b) The kth term in the sequence is 1 .

100
Find the value of k.

Answer (b) k = ................................ [2]

(c) Given that the mth term of the sequence is less than 0.0064, find the smallest value
of m.

Answer (c) m = ............................... [2]

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8 In a group of 8 students there are 5 boys and 3 girls.

Two students are chosen at random.
The tree diagram shows the possible outcomes and their probabilities.

Answer (a)
First student Second student

4 Boy
7
Boy
5
8 3
7 Girl

Boy
3 ......
8
Girl
Girl
......

(a) Complete the tree diagram. [1]

(b) Expressing each answer as a fraction in its lowest terms, find the probability that

(i) two boys are chosen,

Answer (b)(i) ...................................[1]

(ii) at least one boy is chosen.

Answer (b)(ii) ..................................[2]

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9 (a) Write down, in terms of n, an expression for the nth term of the sequence

19 16 13 10 ........... .

Answer (a) ..................................... [2]

(b) Factorise completely

(i) 4x2 – 25y2,

Answer (b)(i) .......................................... [1]

(ii) 5ax – 5a2 – 2x + 2a.

Answer (b)(ii) .......................................... [2]

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10 Look at this pattern.

22 – 02 = 4×1

32 – 12 = 4×2

42 – 22 = 4×3

52 – 32 = 4×4

(a) Write down the 7th line of the pattern.

Answer (a) ........................................................................................................................... [1]

(b) Write down the n th line of the pattern.

Answer (b) ........................................................................................................................... [1]

(c) Use the pattern to find 5212 – 5192 .

Answer (c) ....................................... [1]

(d) Use the pattern to find the positive integers x and y such that x2 – y2 = 484 .

Answer (d) x = .................................

y = ................................. [1]
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11 The first four terms, u1, u2, u3 and u4, in a sequence of numbers are given by

u1 = 1 × 2 + 32 = 11
u2 = 2 × 3 + 42 = 22
u3 = 3 × 4 + 52 = 37
u4 = 4 × 5 + 62 = 56.

(a) Evaluate u5.

Answer ........................................ [1]

(b) Write down an expression, in terms of n, for the nth term, un , of the sequence.

Answer ......................................................................... [1]

(c) Given that un = An2 + Bn + C, find the values of A, B and C.

Answer A = .......................................

B = .......................................

C = .................................. [2]
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12 A series of shapes, made of matchsticks, is shown below.

Shape 1 Shape 2 Shape 3 Shape 4

(a) Draw Shape 4. [1]

(b) The table shows the numbers of matchsticks used to make Shapes 1 and 2.

Shape 1 2 3 4

Number of
12 18
matchsticks

Complete the table for Shapes 3 and 4. [1]

(c) Find an expression, in terms of n, for the number of matchsticks used to make Shape n.

Answer ....................................... [1]

(d) Explain why there is not a shape that is made of 100 matchsticks.

Answer ..............................................................................................................................................

............................................................................................................................................................

...................................................................................................................................................... [1]
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13 The first four terms of a sequence are 55, 53, 49, 41.
The nth term of this sequence is 57 – 2n .

(a) Calculate the fifth term.

Answer ....................................... [1]

(b) Write down the nth term of the sequence 56, 55, 52, 45 ... .

Answer ....................................... [1]

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14 A sequence of diagrams is made using black and white counters.

Diagram 1 Diagram 2 Diagram 3 Diagram 4

The number of black and white counters in each diagram is shown in the table below.

Diagram number 1 2 3 4 5 6

Number of white 1 4 9 16
counters

Number of black 0 2 6 12
counters

(a) Complete the table for Diagrams 5 and 6. [1]

(b) Write an expression, in terms of n, for the number of white counters in the nth diagram.

Answer ..................................... [1]

(c) By considering the number patterns in the table, write an expression, in terms of n, for the
number of black counters in the nth diagram.

Answer ..................................... [1]

(d) What is the total number of counters in the 20th diagram?

Answer ..................................... [1]

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15 The diagrams below show the first three patterns in a sequence.

Pattern 1 Pattern 2 Pattern 3

(a) Complete the table.

Pattern number 1 2 3 4 5
Number of dots 5 8
[1]

(b) Find an expression, in terms of n, for the number of dots in Pattern n.

Answer ..................................... [1]

(c) In this sequence, Pattern p has 83 dots.

Find the value of p.

Answer p = .............................. [2]

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16 The sequence of positive integers is arranged in the pattern below.

Row 1 1 2 3
Row 2 4 5 6
Row 3 7 8 9
Row 4 10 11 12

. . . .
. . . .
. . . .

Row n ............... 3n – 1 ...............

(a) Complete Row n. [1]

(b) The table shows some results obtained from this pattern.

Row number 1 2 3 4 n
Square of the middle number
4 25 64 x
in the row
Product of the first and the last
3 24 63 y
number in the row

(i) Complete the column for Row number 4. [1]

(ii) Find an expression, in terms of n, for y.

Answer ..................................... [1]

(iii) Show that x – y is always equal to 1.

[2]
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17 The n th term of a sequence is 9n + 4.

(a) Calculate the value of the term that is closest to 2012.

Answer ..................................... [2]

(b) Calculate the difference between the 10 th term and the 6 th term.

Answer ..................................... [1]

(c) (i) Find an expression, in terms of x and y, for the difference between the x th term and the
y th term.

Answer ..................................... [1]

(ii) Hence explain why it is not possible for any two terms of this sequence to differ by 123.

Answer ...............................................................................................................................

...................................................................................................................................... [1]
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18 The first and second terms of a sequence are 15 and 11 respectively. For
Examiner’s
B
The nth term of the sequence is 10 + An + . Use
n

(a) Show that A + B = 5 and 4A + B = 2.

[2]

(b) Solve the simultaneous equations.

A+B=5
4A + B = 2

Answer A = ..........................................

B = ......................................... [2]

(c) Hence find the third term of the sequence.

Answer ................................................ [1]

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19 The first four lines of a pattern of numbers are shown below.

1st line 32 − 12 = 8×1

2nd line 52 − 12 = 8 × (1 + 2)

3rd line 72 − 12 = 8 × (1 + 2 + 3)

4th line 92 − 12 = 8 × (1 + 2 + 3 + 4)

(a) Write down the 7th line of the pattern.

Answer ................................................................................................................................ [1]

(b) Write down an expression, in terms of n, to complete the nth line of the pattern.

Answer ............................................................................... = 8 × (1 + 2 + 3 + 4 + … + n) [1]

n (n + 1)
(c) Using the nth line of the pattern, show that 1 + 2 + 3 + 4 + … + n = .
2

[2]
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20 (a) Here are the first four terms of a sequence.

7 11 15 19

Write down an expression, in terms of n, for the nth term of this sequence.

Answer .......................................................... [1]

(b) un is the nth term of another sequence.

Here is the formula connecting the nth and ^n + 1hth terms of this sequence.

3un - 4 = un + 1

The value of u3 is 11.

Find u2 and u4.

Answer u2 = ..................................................

u4 = .................................................. [2]
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21 (a) The term-to-term rule for a sequence is

multiply the previous term by 3 and subtract 1 .

The first three terms in this sequence are 1, 2 and 5 .

Write down the next two terms in this sequence.

Answer ................... , ................... [1]

(b) The nth term of a second sequence is given by the expression 4n - 3 .

Find the number in this sequence that is closest to 150 .

Answer ......................................... [1]

(c) The nth term of a different sequence is given by the expression n2 + 1 .

(i) Write down the first four terms of this sequence.

Answer ........... , ........... , ........... , ........... [1]

(ii) Hence write down an expression, in terms of n, for the nth term of the following sequence.

0 3 8 15 .....

Answer ......................................... [1]

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22 The sequence of diagrams below shows small black and small white squares in an arrangement to
form large squares.

Diagram 1 Diagram 2 Diagram 3

The table below shows the numbers of black and white squares in each diagram.

Diagram (n) 1 2 3 4
Black squares 5 13 25
White squares 4 12 24

Total number of black and

9 25 49
white squares

(a) For each diagram, how many more black squares are there than white squares?

Answer ............................................. [1]

(b) On the table, complete the column for Diagram 4. [1]

(c) Write down an expression, in terms of n, for the total number of black and white squares in
Diagram n.

Answer ............................................. [1]

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23 A pattern of numbers is given below.

1 1 1
Row 1 = -
1#2 1 2

1 1 1
Row 2 = -
2#3 2 3

1 1 1
Row 3 = -
3#4 3 4

1 1 1
Row 4 = -
4#5 4 5
(a) Write down Row 10.

Answer ........................................................................................................................................... [1]

1 1 1 1 2
(b) Adding the first two rows gives the result + = - = .
1#2 2#3 1 3 3
1 1 1 1 1 3
Adding the first three rows gives the result + + = - = .
1#2 2#3 3#4 1 4 4
(i) Write down the result of adding the first four rows.

Answer ................................................................................................................................. [1]

(ii) Use the pattern to write down

1 1 1 1
(a) the value of + + + ... + ,
1#2 2#3 3#4 19 # 20

Answer .............................................. [1]

109
(b) the number of rows that add up to ,
110

Answer .............................................. [1]

(c) an expression, in terms of n, for the result of adding the first n rows.

Answer .............................................. [1]

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24 (a) Thefirstfourtermsofasequence,S,are89,83,77,71.

(i) FindanexpressionforSn,thenthtermofthissequence.

AnswerSn=.................................... [2]

(ii) FindthesmallestvalueofnforwhichSn<0.

Answern=...................................... [1]

(b) Thenthtermofadifferentsequence,T,isgivenbyTn = n 2 - 4n .

(i) FindandsimplifyanexpressionforTn + 1 - Tn .

Answer............................................ [2]

(ii) ThedifferencebetweenTp + 1 andTp is75.

Findthevalueofp.

Answerp=..................................... [1]
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25 A sequence of patterns is made using dots and

lines.

Pattern 1 Pattern 2 Pattern 3

Pattern number ( p) 1 2 3 4
Number of dots (d ) 6 11 16

(a) Complete the table for Pattern 4. [1]

(b) Find a formula for the number of dots, d, in Pattern p.

Answer d = .................................. [2]

26 The first term of a sequence is 13.

The following terms are found by alternately adding 4 and 6 to the previous term.
The first six terms are

13 17 23 27 33 37

(a) Write down the next two terms of the sequence.

Answer ................................................. [1]

(b) Write down the value of the term that is closest to 999.

Answer ................................................. [1]

(c) Write down the difference between the values of the 91st and 93rd terms.

Answer ................................................. [1]

(d) Find the 80th term.

Answer ................................................. [1]

(e) The nth term is 203.

Find n.

Answer n = ................................................. [1]

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27 A sequence of diagrams is made using

counters.

Diagram 1 Diagram 2 Diagram 3 Diagram 4

(a) Complete the table.

Diagram number 1 2 3 4 5

Number of counters 4 6 8
[1]

(b) Find an expression, in terms of n, for the number of counters in Diagram n.

Answer ............................................. [1]

(c) In this sequence, Diagram p has 200 counters.

Find the value of p.

Answer p = ............................................. [2]

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28 (a) The nth term of a sequence is given by n2 - 5n.

(i) Find the 2nd term in the sequence.

Answer ........................................... [1]

(ii) The pth term in the sequence is 150.

Find the value of p.

Answer p = ..................................... [2]

(b) The nth term of another sequence is given by 3n 2 - kn.

The 5th term in this sequence is 55.

Find the value of k.

Answer k = ..................................... [2]

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29

Triangle 1 Triangle 2 Triangle 3 Triangle 4

The diagrams show a sequence of triangles made up of identical sticks.

Each triangle has two more sticks on each edge than its previous triangle.
The table shows information relating to this sequence.

Triangle number 1 2 3 4 n

Number of sticks on
1 3 5 x
each side
Number of sticks in
3 9 15 y
the triangle

(a) Complete the column for triangle 4. [1]

(b) Find an expression, in terms of n, for x.

Answer x = .................................... [1]

(c) Find an expression, in terms of n, for y.

Answer y = .................................... [1]

(d) The total number of sticks in the first triangle = 3

The total number of sticks in the first two triangles = 12
The total number of sticks in the first three triangles = 27

(i) Write down the total number of sticks in the first four triangles.

Answer ......................................... [1]

(ii) Find an expression, in terms of n, for the total number of sticks in the first n triangles.

Answer ......................................... [2]

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30 Given that 192 # 64.3 = 12 345.6 , write down the values of

(a) 0.192 # 643 ,

Answer ......................................... [1]

12.3456
(b) .
192
Answer .......................................... [1]

31 Two sequences have 1, 3, 5 as their first three terms.

(a) In the first sequence, each term is 2 more than the term before it.

(i) Find an expression, in terms of n, for the nth term.

Answer .......................................... [1]

(ii) The kth term of this sequence is 841.

Find the value of k.

Answer k = .................................... [1]

(b) The nth term of the second sequence is

( n - 1) ( n - 4)
2n - 1 - .
2
(i) Find the fourth term of this sequence.

Answer .......................................... [1]

(ii) Find the fifth term of this sequence.

Answer .......................................... [1]