What is covered in Grade 10 Mathematics Number Patterns?

Investigate linear number patterns (constant first difference). Derive the general term Tₙ = T₁ + (n−1)d from first principles. Find any term or the position of a given value.

Is Grade 10 Number Patterns tested in Paper 1 or Paper 2?

Grade 10 Mathematics Number Patterns is assessed in Paper 1 of the end-of-year examinations, as per the CAPS Mathematics curriculum guidelines.

When is Number Patterns taught in Grade 10 Mathematics?

Number Patterns is taught in Term 4 of Grade 10 according to the CAPS Annual Teaching Plan (ATP) for Mathematics in South Africa.

Is Grade 10 Mathematics Number Patterns on the CAPS curriculum?

Yes. Number Patterns is part of the official CAPS Mathematics curriculum for Grade 10, Term 4 in South Africa. The content on MathSciBuddy follows the Annual Teaching Plan (ATP) set by the Department of Basic Education.

Is this Grade 10 Mathematics study material free?

Yes. All chapter notes, definitions, formulas, and worked examples on MathSciBuddy are completely free to read. No account or subscription is needed to access CAPS-aligned study notes.

Grade 10 Mathematics

Grade 10 · Term 4Mathematics

Number Patterns

We investigate number sequences where consecutive terms differ by the same constant amount (linear patterns). We find the general term of such a sequence and use it to calculate any term or find the position of a given value.

Week 3

12.1 Linear Number Patterns

Investigate number patterns with a constant difference between consecutive terms
Determine the general term of a linear pattern (without using a formula — derive it from first principles)
Use the general term to calculate specific terms and find the position of a given term

🌍

Real-World Connection

A taxi charges R8 per kilometre. After 1 km: R8; after 2 km: R16; after 3 km: R24. The differences are all R8 — this is a linear pattern. The general term T_n = 8n lets you instantly calculate the fare for any distance without listing all the values.

Definition

Constant Difference (Linear Pattern)

A number sequence has a constant (first) difference if each term is obtained by adding the same fixed value d to the previous term. The sequence increases if d > 0 and decreases if d < 0.

T_2 - T_1 = T_3 - T_2 = T_4 - T_3 = d\quad(\text{constant})

Property / Rule

General Term of a Linear Pattern

For a linear pattern with first term T₁ and constant difference d, the general term (n-th term) is derived by observing how each term is built from T₁ and d. The pattern adds d exactly (n−1) times to reach the n-th term.

T_n = T_1 + (n-1)d

💡 Tip

Always verify your general term by testing it for n = 1, n = 2, and n = 3. The formula should reproduce the given terms exactly. When asked to find the POSITION of a term, set Tₙ = given value and solve for n — the answer must be a positive integer.

ℹ️ Note

The general term Tₙ = T₁ + (n−1)d can be simplified to Tₙ = an + b form (where a = d and b = T₁ − d). Both forms are correct. The ATP requires you to derive the formula rather than simply quote it.

Worked Examples

Worked Example

Find the general term from the pattern

Problem

T_{20}

Worked Example

Find which term has a given value

Problem

For the pattern 7; 12; 17; 22; \dots — is 102 a term? If so, which term?

Worked Example

Decreasing linear pattern

Problem

A pattern begins: 50; 45; 40; 35; \dots Find the general term and determine which term is the first negative term.

Activity — 8 Questions

CAPS Cognitive Level Distribution

L1 · Knowledge2 Q

L2 · Routine Procedures2 Q

L3 · Complex Procedures2 Q

L4 · Problem Solving2 Q

L1 · Knowledge1 mark

State the constant difference for: 3; 7; 11; 15; …

L1 · Knowledge2 marks

Find the 5th and 6th terms of the pattern: 2; 9; 16; 23; …

L2 · Routine Procedures3 marks

Find the general term of: 4; 10; 16; 22; …

L2 · Routine Procedures3 marks

Use the general term

T_n = 5n - 3

to find

T_{15}

and the position of the term equal to 97.

L3 · Complex Procedures5 marks

A pattern has

T_3 = 17

and

T_7 = 33

. Find the general term and

T_{50}

L3 · Complex Procedures4 marks

For the pattern

-2;\;3;\;8;\;13;\;\ldots

, find the first term greater than 100.

L4 · Problem Solving5 marks

Sequence A:

3;\;9;\;15;\;21;\;\ldots

and Sequence B:

5;\;9;\;13;\;17;\;21;\;\ldots

share the term 9. Find the NEXT value that appears in both sequences.

L4 · Problem Solving4 marks

Determine whether 500 is a term of the pattern: 3; 10; 17; 24; … Justify your answer.

Measurement