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Patterns & Sequences — Grade 10

Learn to describe number patterns and find the general term of a linear (arithmetic) pattern, Tn = a + (n−1)d. Notes, past papers and a quiz — everything for this topic is one click away.

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Grade 10 CAPS Mathematics

Patterns & Sequences

Describing number patterns, and finding the general term of a linear pattern, \(T_n=a+(n-1)d\).

What Is a Number Pattern?

A list of numbers, called terms, that follow a rule.

Terms and positions

We write \(T_1,T_2,T_3,\ldots\) for the terms, and \(T_n\) for the term in position \(n\).

Describing in words

\(3;\;6;\;9;\;12;\;\ldots\) can be described as “start at 3, add 3 each time.”

Describing with a formula

The same pattern, written as a rule: \(T_n=3n\). Substituting \(n=1,2,3,4\) reproduces \(3;6;9;12\).

Not every pattern is a number list

A pattern can also be built from shapes or dots, growing according to a rule — you'll see one on the next slide.

Watch the wording
A pattern built from shapes or dots is sometimes called a “geometric pattern”. That is not the same as a “geometric sequence” (a number pattern with a constant ratio) — that's different, Grade 12 content. Don't mix the two up.

Two Kinds of Rule

“Add 4” and “\(T_n=4n-1\)” are both correct — but they don't do the same job.

Term-to-term rule

For \(3;\;7;\;11;\;15;\;\ldots\): “start at 3, add 4 each time.” This tells you how to get from one term to the next — but to find \(T_{50}\) this way, you'd have to add 4 forty-nine times.

Position-to-term rule

The same pattern as \(T_n=4n-1\). This tells you the value at any position directly — substitute \(n=50\) and you're done, no counting required.

Why this matters
Every worked example in this lesson is about turning a term-to-term description into a position-to-term rule — that's the actual skill CAPS is testing, not just spotting “add 4.”
Worked example — from a diagram to a table Level 1

Count the dots in each pattern, then build a table of position vs. value.

Three L-shaped dot patterns with 3, 5 and 7 dots Three L-shaped arrangements of dots: pattern 1 has 3 dots, pattern 2 has 5 dots, pattern 3 has 7 dots, each formed by a growing bottom row and left column of dots. Pattern 1 Pattern 2 Pattern 3
Show solution
Position \(n\)123
Term value \(T_n\)357
First difference—22

The first difference is constant \((d=2)\), so this is a linear pattern with \(a=3\).

Worked example — read from a diagram Level 1

The diagram shows squares joined edge to edge, built from matchsticks. Count the matchsticks in each panel, then describe the pattern.

Three panels of squares joined in a row, built from matchsticks Panel one shows a single square made of 4 matchsticks. Panel two shows two squares joined in a row made of 7 matchsticks. Panel three shows three squares joined in a row made of 10 matchsticks. T₁=4 T₂=7 T₃=10
Show solution
  1. 1Counting matchsticks: \(T_1=4\), \(T_2=7\), \(T_3=10\).
  2. 2Each new square shares one side with the square before it, so it only needs \(3\) new matchsticks: the pattern is “start at 4, add 3 each time.”
  3. 3First differences \(3;3\) are constant, so this is a linear pattern with \(d=3\).
Quick Check

A pattern has a constant first difference. This means it is:

For the pattern \(4;\;7;\;10;\;13;\ldots\), the common difference \(d\) is:

Which of these is a position-to-term rule for \(5;\;9;\;13;\;17;\ldots\)?

Finding Any Term Without Counting

Reason it out from \(a\) and \(d\) — don't just memorise a formula.

Build the pattern, term by term
\(T_1=a\). No \(d\) has been added yet.
\(T_2=a+d\). One \(d\) has been added.
\(T_3=a+2d\). Two \(d\)'s have been added.
The number of \(d\)'s added is always one less than the term number — so for \(T_n\), you add \(d\) exactly \((n-1)\) times.
Written as a rule
Continuing this reasoning for any term gives \(T_n=a+(n-1)d\). Use it as a shortcut once you understand why it works — the reasoning above, not this line, is what CAPS expects you to be able to reproduce at this level. (The formal “arithmetic sequence” formula and its name are Grade 12 content.)
Worked example — a non-whole-number \(d\) Level 1

A pattern begins \(3;\;4.5;\;6;\;7.5;\;\ldots\) Determine \(d\), and find the next term.

Show solution
  1. 1\(d=T_2-T_1=4.5-3=1.5\). Check: \(T_3-T_2=6-4.5=1.5\) too — constant.
  2. 2\(a=3\), so \(T_n=3+(n-1)(1.5)\).
  3. 3The next term is \(T_5=7.5+1.5=\boxed{9}\).
Common mistake
Don't assume \(d\) has to be a whole number — it can be a fraction or a decimal, as long as it's the same value every time.
Worked example — increasing pattern Level 1

For the pattern \(2;\;5;\;8;\;11;\;\ldots\), find \(T_n\), then determine \(T_{10}\).

Show solution
  1. 1Common difference: \(d=T_2-T_1=5-2=3\).
  2. 2First term: \(a=2\).
  3. 3\(T_n=a+(n-1)d=2+(n-1)(3)=2+3n-3=\boxed{3n-1}\).
  4. 4\(T_{10}=3(10)-1=\boxed{29}\).
Worked example — decreasing pattern Level 2

For the pattern \(8;\;5;\;2;\;-1;\;\ldots\), find \(T_n\), then determine \(T_{10}\).

Show solution
  1. 1Common difference: \(d=T_2-T_1=5-8=-3\).
  2. 2First term: \(a=8\).
  3. 3\(T_n=a+(n-1)d=8+(n-1)(-3)=8-3n+3=\boxed{11-3n}\).
  4. 4\(T_{10}=11-3(10)=11-30=\boxed{-19}\).
Worked example — error analysis Level 1

A learner claims the pattern \(5;\;8;\;11;\;14;\;\ldots\) has general term \(T_n=5+3n\). Show that this is wrong, then find the correct rule.

Show solution
  1. 1Test the claimed rule at \(n=1\): \(T_1=5+3(1)=8\). But the actual first term is \(5\), not \(8\) — the rule is wrong.
  2. 2The learner forgot the \((n-1)\): they wrote \(a+dn\) instead of \(a+(n-1)d\).
  3. 3Correct working: \(d=3\), \(a=5\), so \(T_n=5+3(n-1)=\boxed{3n+2}\).
  4. 4Check: \(T_1=3+2=5\) ✓, \(T_4=12+2=14\) ✓.
Worked example — two rules, same pattern Level 1

For \(7;\;12;\;17;\;22;\;\ldots\), Learner A writes \(T_n=7+5(n-1)\) and Learner B writes \(T_n=5n+2\). Who is correct?

Show solution
  1. 1Expand Learner A's version: \(7+5(n-1)=7+5n-5=5n+2\).
  2. 2That's exactly Learner B's rule — both are correct, just written in different forms.
  3. 3Check either version at \(n=1\): \(5(1)+2=7\) ✓.
Why this matters
Before deciding a rule is “wrong,” expand it and compare — two rules that look different can be algebraically identical.
Worked example — filling in the gaps Level 1

A linear pattern has \(T_1=5\) and \(T_4=20\). Determine the missing terms \(T_2\) and \(T_3\).

Show solution
  1. 1Going from \(T_1\) to \(T_4\) is \(3\) steps of \(d\): \(T_4=T_1+3d\).
  2. 2\(20=5+3d \Rightarrow 15=3d \Rightarrow d=5\).
  3. 3\(T_2=5+5=\boxed{10}\), \(T_3=10+5=\boxed{15}\).
  4. 4Check: \(T_4=15+5=20\) ✓.
Worked example — a decreasing pattern Level 3

A linear pattern has \(T_1=100\) and \(T_5=60\). Determine the missing terms \(T_2\), \(T_3\) and \(T_4\).

Show solution
  1. 1Going from \(T_1\) to \(T_5\) is \(4\) steps of \(d\): \(T_5=T_1+4d\).
  2. 2\(60=100+4d\Rightarrow-40=4d\Rightarrow d=-10\)
  3. 3\(T_2=100-10=\boxed{90}\), \(T_3=90-10=\boxed{80}\), \(T_4=80-10=\boxed{70}\)
  4. 4Check: \(T_5=70-10=60\) ✓
Quick Check

A learner tests \(T_n=2+3n\) against the pattern \(2;\;5;\;8;\ldots\) At \(n=1\) this gives \(T_1=5\), not \(2\). What is the most likely error?

A linear pattern has \(T_1=6\) and \(T_5=26\). What is \(d\)?

Quick Check

In \(T_n=a+(n-1)d\), what does \(a\) represent?

For \(a=2\), \(d=5\), what is \(T_6\)?

Worked example — which term is it? Level 1

A pattern begins \(3;\;7;\;11;\;15;\;\ldots\) Which term equals \(99\)?

Show solution
  1. 1\(d=4\), \(a=3\), so \(T_n=3+(n-1)(4)=4n-1\)
  2. 2Set \(T_n=99\): \(4n-1=99\Rightarrow4n=100\Rightarrow n=25\)
  3. 3\(\boxed{99\text{ is the 25th term}}\)
Worked example — is it even a term? Level 3

A pattern begins \(4;\;9;\;14;\;19;\;\ldots\) (a) Which term equals \(149\)? (b) Is \(152\) a term of this pattern?

Show solution
  1. 1\(d=5\), \(a=4\), so \(T_n=4+(n-1)(5)=5n-1\).
  2. 2(a) Set \(T_n=149\): \(5n-1=149 \Rightarrow 5n=150 \Rightarrow n=\boxed{30}\). So \(149\) is the 30th term.
  3. 3(b) Set \(T_n=152\): \(5n-1=152 \Rightarrow 5n=153 \Rightarrow n=30.6\).
  4. 4\(n\) must be a positive whole number (a term position), and \(30.6\) is not — so \(\boxed{152\text{ is not a term}}\) of this pattern.

Patterns and Graphs: Points on a Line

Plotting position against value turns a linear pattern into a straight line.

The points (1,4), (2,9), (3,14) and (4,19) lying on a straight line Four points plotted with position n on the horizontal axis and term value Tn on the vertical axis, all lying exactly on a straight line, for the pattern 4, 9, 14, 19. n Tₙ (1,4) (2,9) (3,14) (4,19)
Reading the diagram: the pattern \(4;9;14;19;\ldots\) has \(T_n=5n-1\). Plotting \((n,T_n)\) for each position gives points that fall exactly on a straight line — that's what “linear” means.
Why a straight line?

\(T_n=5n-1\) has the same structure as a straight-line equation \(y=mx+c\), with \(n\) playing the role of \(x\) and \(d\) playing the role of the gradient \(m\).

The dashed line isn't part of the pattern

It's drawn to help you see the trend. Only the marked points at whole-number positions \((n=1,2,3,\ldots)\) are actual terms — there's no “\(T_{2.5}\).”

Worked example — a savings pattern Level 1

Sam starts a savings jar with R50, then adds R15 to it every week. Let \(T_n\) be the total amount saved after \(n\) weeks. Determine \(T_n\), then find the total after 20 weeks.

Show solution
  1. 1After week 1: \(T_1=50+15=65\). Each extra week adds another R15, so \(d=15\).
  2. 2\(T_n=65+(n-1)(15)=65+15n-15=\boxed{15n+50}\).
  3. 3\(T_{20}=15(20)+50=300+50=\boxed{R350}\).
Worked example — a real-life pattern Level 3

A phone plan costs R150 to connect, plus R2 for every minute of calls made. Let \(T_n\) be the total cost after \(n\) minutes of calls. Determine \(T_n\), then find the cost after 45 minutes.

Show solution
  1. 1After 1 minute: \(T_1=150+2(1)=152\). Each extra minute adds another R2, so \(d=2\).
  2. 2\(T_n=152+(n-1)(2)=152+2n-2=\boxed{150+2n}\).
  3. 3\(T_{45}=150+2(45)=150+90=\boxed{R240}\).
Quick Check

Solving \(T_n=k\) for \(n\) gives a non-integer answer. This means:

For \(T_n=6n-2\), which term equals \(40\)?

Putting It Together

Every Grade 10 pattern question follows this process.

StepWhat to do
1List or count the terms (from a list, a table, or a diagram). Find the first difference between consecutive terms.
2If the first difference is constant, that's \(d\), and the pattern is linear — \(d\) can be negative or a fraction.
3Reason out \(T_n=a+(n-1)d\), using \(a=T_1\). Before trusting any proposed rule, test it at \(n=1\).
4To find a term, substitute \(n\). To find which term equals a value, solve for \(n\) — and reject any answer that isn't a positive whole number.
5Plotting \((n,T_n)\) always gives points on a straight line for a linear pattern — only whole-number positions are real terms.
Looking ahead
Not every pattern has a constant first difference. Next year, Grade 11 shows you what to do when the first differences change but the second differences stay constant — that's a quadratic pattern, and the same position-to-term thinking you practised here is exactly what makes it possible.
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What CAPS Expects You to Know

The Grade 10 Patterns knowledge statement this page is built from.

  1. 1

    Investigate number patterns leading to those where there is a constant difference between consecutive terms, and the general term (without using a formula) is therefore linear.

How to Use This Lesson

A few practical notes before you start.

  • Always find the first difference between consecutive terms before writing any formula.
  • \(a\) is always the first term, never the common difference — a very common mix-up.
  • If a value doesn't give a whole-number, positive \(n\) when you solve for it, it simply isn't a term in the pattern.
  • Try each worked example yourself before pressing “Show solution.”

Learn It in Short Videos

Four free, independent videos covering the core Grade 10 skills — not made by Equation Station SA.

What Is a Linear Pattern?

A quick, clear introduction to sequences with a constant common difference.

Patrick JMT · Quick Intro to Arithmetic Sequences

Reading a Pattern From a Table

Turning a table of values into a rule — the same skill as the diagram-to-table worked example above.

Khan Academy · Math patterns example 1

Finding the General Term

Using \(T_n=a+(n-1)d\) to find any term of a linear pattern.

Khan Academy · Using arithmetic sequences formulas

Patterns and Graphs

Plotting a sequence's terms as points — the same idea as the “Patterns and Graphs” slide above.

Khan Academy · Number patterns: visualizing sequence relationships

Common Exam Mistakes

Avoid these errors. They cost marks every year.

Swapping \(a\) and \(d\)

Using the common difference where the first term should go, or the other way around.

Forgetting the \((n-1)\)

Writing \(T_n=a+nd\) instead of \(T_n=a+(n-1)d\) — check it against \(T_1\): substituting \(n=1\) must give exactly \(a\).

Accepting a non-integer \(n\)

Solving for \(n\) and getting a decimal, but still writing an answer such as “the 30.6th term.”

Confusing "geometric pattern" and "geometric sequence"

A geometric pattern (shapes/dots) has nothing to do with a geometric sequence (constant ratio) — they're unrelated ideas that just share a word.

Trusting a rule after checking only one term

A rule can match \(T_1\) by coincidence and still be wrong — always check it against at least two terms before accepting it.

Reading in-between values off the graph

The line joining \((n,T_n)\) points is a visual aid only — there is no real \(T_{2.5}\); only whole-number positions are actual terms.

This Topic in Other Grades

Patterns & Sequences builds across all three FET grades — revise or preview the other levels here.

Grade 11

Quadratic number patterns, general term \(T_n=an^2+bn+c\)

Study this

Grade 12

Arithmetic and geometric series, sigma notation, sum to infinity

Study this

Practise This Topic

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Past Papers
Patterns & Sequences Grade 10 Past Papers

Exam-style Grade 10 number pattern questions arranged by cognitive level, with real citations from the DBE/provincial archive.

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Student Questions

Frequently Asked Questions

Straight answers to common Grade 10 CAPS questions about number patterns.

What does Grade 10 Patterns & Sequences cover?

Describing number patterns in words and as a formula, and finding the general term of a linear (arithmetic) pattern: sequences with a constant first difference, Tn = a+(n-1)d.

What is a common difference?

The constant amount added (or subtracted) to get from one term to the next in a linear pattern. It is written d, and calculated as d = T2 - T1 (or any Tn - T(n-1)).

Does Grade 10 cover quadratic or geometric patterns?

No. Grade 10 focuses only on linear patterns with a constant first difference. Quadratic number patterns (constant second difference) are Grade 11 content, and geometric sequences (constant ratio) are Grade 12 content.

Is a "geometric pattern" the same as a "geometric sequence"?

No. A pattern built from shapes or dots (like a growing row of tiles) is sometimes called a geometric pattern, but that is not the same as a "geometric sequence" (a number sequence with a constant ratio, which is Grade 12 content). The similar names can cause confusion.

Which resource should I open first for this topic?

Everything you need to learn the topic is on this page already. Once you've been through the notes above, work through the Past Question Papers, then finish with the Test Your Knowledge quiz as a self-check.