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Exam-Style Questions.

Problems adapted from questions set for previous Mathematics exams.

1.

GCSE Higher

The first five terms of an arithmetic sequence are:

$$-20, -17, -14, -11, -8$$

Find an expression for the \(n^{th}\) term of the sequence.


2.

GCSE Higher

Here is a picture of four models. Some of the cubes are hidden behind other cubes.

cube models

Model one consists of one cube. Model two consists of four cubes and so on.

(a) How many cubes are in the third model?

(b) How many cubes are in the fourth model?

(c) If a fifth model were built, how many cubes would it take?

(d) Find an expression for the number of cubes used in the nth model.

(e) Sketch a side view, front view and plan view of the fourth model.


3.

GCSE Higher

(a) Find the \(n\)th term of the sequence 7, 13, 19, 25,...

(b) In a sequence of four numbers, the difference between each number is 9.
The sum of the four numbers is 2.
What are the numbers in the sequence?
You must show all your working.


4.

GCSE Higher

Work out an expression for the nth term of the quadratic sequence

$$0,9,22,39,60, ...$$

Give your answer in the form \(an^2 + bn + c\) where a, b and c are constants.


5.

GCSE Higher

Find an expression, in terms of \(n\), for the \(n\)th term of the sequence that has the following first five terms:

$$6 \qquad 13 \qquad 23 \qquad 36 \qquad 52 $$

6.

IGCSE Extended
Triangle Fractal

The diagrams above show a growing fractal of triangles. The sides of the largest equilateral triangle in each diagram are of length 1 metre.

In the second diagram there are four triangles each with sides of length \(\frac{1}{2}\) metre.

In the third diagram there are 16 triangles each with sides of length \(\frac{1}{4}\) metre.

(a) Complete this table for more diagrams.

Diagram 1 Diagram 2 Diagram 3 Diagram 4 Diagram 5 Diagram 6   Diagram \(n\)
Length of Side 1 \(\frac{1}{2}\) \(\frac{1}{4}\)  
Power of 2 20 2-1 2-2  

(b) Complete this table for the number of the smallest triangles in diagrams 4, 5 and 6.

Diagram 1 Diagram 2 Diagram 3 Diagram 4 Diagram 5 Diagram 6   Diagram \(n\)
Number of smallest triangles 1 4 16
Power of 2 20 22 24

(c) Calculate the number of the smallest triangles in the diagram where the smallest triangles have sides of length \(\frac{1}{256}\) metre.


7.

GCSE Higher

The diagrams below show a sequence of patterns made from red and yellow tiles.

Pattern

(a) Find an expression, in terms of \(n\), for the number of red tiles in pattern \(n\).

The total number of red and yellow tiles in each pattern is always the sum of the squares of two consecutive whole numbers.

(b) Find an expression, in terms of \(n\), for the total number tiles in Pattern \(n\) giving your answer in its simplest form.

(c) Is there a pattern for which the total number of tiles is 303? Give a reason for your answer.

(d) Explain why the total number of tiles in any pattern of this sequence is always an odd number.


8.

GCSE Higher

(a) A sequence is defined by the following rule where \(u_n\) is the \(n^{th}\) term of the sequence:

$$u_{n+1}=u_n^2-5u_n+21$$

If \(u_1=3\) find \(u_2\) and \(u_3\).

(b) A different sequence is defined by the following rule:

$$u_{n+1}=u_n^2-8u_n+17$$

If \(u_1=5\) find \(u_2\) and \(u_3\) and \(u_{50}\).


The exam-style questions appearing on this site are based on those set in previous examinations (or sample assessment papers for future examinations) by the major examination boards. The wording, diagrams and figures used in these questions have been changed from the originals so that students can have fresh, relevant problem solving practice even if they have previously worked through the related exam paper.

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