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Number patterns and sequences

27 marks in our Paper 1 questions. 8 questions from 2024.

Practise this topic, marked

This topic covers number patterns, arithmetic and geometric sequences, and series written in sigma notation, including patterns drawn from shapes. The 8 questions here are worth 27 marks in the November 2024 Paper 1 set. Of the 27 marks, 15 are for the correct final answer and 12 for method and working.

The 2024 Diagnostic Report says candidates do not see that a linear pattern given in a question is the first difference of a quadratic pattern. In a circle question they add radii when the question asks for the sum of the areas, whose common ratio is not the ratio of the radii. In sigma notation they do not write the sum from n equal to 1 to the right upper limit, so it does not equal the total given. Decide which sequence you are summing before you pick a formula.

Where marks are lost: in our words, from Department of Basic Education, NSC 2024 Diagnostic Report, Part 1 (Book 1).

Questions

These questions are not yet checked by a teacher.

Look at the product

(1 + \frac{1}{3}) (1 + \frac{1}{4}) (1 + \frac{1}{5}) \dots (1 + \frac{1}{n})

n \geq 3

and the last factor is

1 + \frac{1}{n}

. Find ALL the values of

n

for which this product is a whole number.

(3)Based on Nov 2024 · Paper 1 · Q1.3

An arithmetic series has a first term of

4

and a common difference of

3

. The series has

30

terms. Calculate the sum of all

30

(2)Based on Nov 2024 · Paper 1 · Q2.1.1

The first term of an arithmetic series is

6

and the common difference is

4

. The series has

15

terms. It is then extended so that it has

60

terms in total, and the sum of all

60

7440

. Write down an equation, using sigma notation, for the sum of only the new terms that were added to the original series.

(4)Based on Nov 2024 · Paper 1 · Q2.1.2

The first differences of a quadratic number pattern are

4; 6; 8; \dots

T_{60} = 3 500

, calculate the value of

T_{59}

(3)Based on Nov 2024 · Paper 1 · Q2.2.1

The first differences of a quadratic number pattern are:

4; 6; 8; \dots

The third term of the pattern is

22

. Determine the general term,

T_{n}

, of the pattern.

(5)Based on Nov 2024 · Paper 1 · Q2.2.2

A circle with radius 20 cm is drawn. A second, smaller circle is drawn so that it passes through the centre of the first circle and touches the first circle from the inside, as shown in the diagram below. A third circle is drawn in the same way: it passes through the centre of the second circle and touches the second circle from the inside. This carries on, so that every new circle has a radius equal to half of the radius of the circle before it. Write down the radius of the 4th circle.

(2)Based on Nov 2024 · Paper 1 · Q3.1

A circle of radius 8 cm is drawn. A second, smaller circle is drawn so that it passes through the centre of the first circle and touches the first circle from the inside. A third circle is drawn in the same way inside the second circle, passing through the centre of the second circle and touching it from the inside. This pattern of circles carries on, as shown in the diagram below. Calculate the sum of the areas of the first 8 circles. Give your answer correct to two decimal places.

(4)Based on Nov 2024 · Paper 1 · Q3.2

A circle of radius 20 cm is drawn. A second circle is drawn so that it passes through the centre of the first circle and touches the first circle from the inside. A third circle is drawn in the same way inside the second circle: it passes through the centre of the second circle and touches the second circle from the inside. The process carries on forever, as shown in the diagram below. Which circle in this pattern has a diameter of

\frac{5}{64}

(4)Based on Nov 2024 · Paper 1 · Q3.3

Updated 12 November 2026