Patterns & Sequences Past Question Papers | Grade 10

How to use this bank.

Start at Level 1 and move up — don't jump to Level 4 first.
Always find the first difference between consecutive terms before writing any formula.
\(a\) is always the first term, never the common difference.
Reveal the memo only after a genuine attempt.

Accuracy note: every question below was independently solved from scratch before publication. Three questions (Q5, Q9, Q10) are sourced from the real DBE/provincial exam archive and were independently re-derived and confirmed to match the official memo before being included; every other question is an original, labelled "Equation Station SA Practice Question," per this site's citation policy of never attaching a specific exam attribution unless it has been confirmed against the archived paper.

L1 — Knowledge 4 Qs

L2 — Routine Procedures 5 Qs

L3 — Complex Procedures 4 Qs

L4 — Problem Solving 4 Qs

20%

Level 1 | Knowledge

Describe and Recall

Find a common difference, recall terms, and recognise when a pattern isn't linear.

Q1Equation Station SA Practice Question1 mark

Common difference

Finding \(d\)

Determine the common difference of the pattern \(5;\;9;\;13;\;17;\;\ldots\)

Memo

✓ \(d=9-5=4\)

Q2Equation Station SA Practice Question2 marks

Common difference

Writing Out Terms

A linear pattern has \(T_1=3\) and \(d=6\). Write down \(T_2\) and \(T_3\).

Memo

✓ \(T_2=3+6=9\)✓ \(T_3=9+6=15\)

Q3Equation Station SA Practice Question2 marks

Classification

Is This Pattern Linear?

Determine whether \(4;\;8;\;13;\;19;\;\ldots\) has a constant first difference.

Memo

✓ First differences: \(4;5;6\) — not constant✓ This is not a linear pattern

Q4Equation Station SA Practice Question2 marks

Description

Describe the Pattern in Words

Describe, in words, the pattern \(10;\;7;\;4;\;1;\;\ldots\)

Memo

✓ Start at 10, subtract 3 each time

35%

Level 2 | Routine Procedures

Finding and Using \(T_n\)

Determine the general term, and use it to find a specific term.

Q5Gauteng, November 20225 marks

General term

Two Equivalent Formulas

For the sequence \(2;\;6;\;10;\;14;\;\ldots\), Letti says \(T_n=4n-2\) and Zain says \(T_n=4(n-1)+2\). Who is right? Use algebra to justify your answer, then determine \(T_{27}\).

Memo

✓ Expand Zain's formula: \(4(n-1)+2=4n-4+2=4n-2\) — identical to Letti's✓ Both are right — they are algebraically the same formula written two different ways✓ \(T_{27}=4(27)-2=108-2=\boxed{106}\)

Q6Equation Station SA Practice Question4 marks

General term

Finding \(T_n\) With a Negative \(d\)

For the pattern \(20;\;15;\;10;\;5;\;\ldots\), determine \(T_n\), then find \(T_{12}\).

Memo

✓ \(d=15-20=-5\), \(a=20\)✓ \(T_n=20+(n-1)(-5)=25-5n\)✓ \(T_{12}=25-60=\boxed{-35}\)

Q7Equation Station SA Practice Question4 marks

General term

Finding \(T_n\)

For the pattern \(6;\;13;\;20;\;27;\;\ldots\), determine \(T_n\), then find \(T_{25}\).

Memo

✓ \(d=7\), \(a=6\)✓ \(T_n=6+(n-1)(7)=7n-1\)✓ \(T_{25}=175-1=\boxed{174}\)

Q8Equation Station SA Practice Question2 marks

General term

Evaluate a Given Term

For \(T_n=7n-3\), determine \(T_9\).

Memo

✓ \(T_9=7(9)-3=63-3=\boxed{60}\)

Q9Eastern Cape, November 20244 marks

Recognising a pattern

Adapting a Known Square Pattern

Given that \(1;\;4;\;9;\;16;\;\ldots\) has general term \(T_n=n^2\), use this to determine the general term for: (a) \(2;\;5;\;10;\;17;\;\ldots\) (b) \(0;\;1;\;4;\;9;\;\ldots\)

Memo

✓ (a) Each term is 1 more than the matching square number: \(\boxed{T_n=n^2+1}\)✓ (b) Each term is the square of the position before it: \(\boxed{T_n=(n-1)^2}\)

30%

Level 3 | Complex Procedures

Solving for \(n\), and Missing Terms

Find a missing term, and use \(T_n\) to answer "which term" or "is this a term" questions.

Q10Eastern Cape, November 20249 marks

General termMissing term

Full Pattern Investigation

Consider the linear pattern \(37;\;33;\;29;\;p;\;21;\;\ldots\) Determine: (a) the value of \(p\) (b) the general term \(T_n\) (c) \(T_{15}\) (d) which term is the first to have a negative value.

Memo

✓ (a) \(d=33-37=-4\), so \(p=29-4=\boxed{25}\)✓ (b) \(T_n=37+(n-1)(-4)=41-4n\)✓ (c) \(T_{15}=41-60=\boxed{-19}\)✓ (d) \(41-4n<0\Rightarrow n>10.25\); \(T_{10}=1\) (positive), \(T_{11}=-3\) (negative) — \(\boxed{T_{11}\text{ is the first negative term}}\)

Q11Equation Station SA Practice Question3 marks

Solving for n

Which Term Equals a Value?

For \(T_n=6n-1\), determine which term equals \(89\).

Memo

✓ \(6n-1=89\Rightarrow6n=90\Rightarrow n=\boxed{15}\)

Q12Equation Station SA Practice Question3 marks

Verification

Is This Value a Term?

Is \(152\) a term of the pattern \(4;\;9;\;14;\;19;\;\ldots\)? Justify your answer.

Memo

✓ \(d=5\), \(a=4\), so \(T_n=5n-1\)✓ \(5n-1=152\Rightarrow5n=153\Rightarrow n=30.6\)✓ \(n\) is not a positive whole number, so \(\boxed{152\text{ is not a term}}\)

Q13Equation Station SA Practice Question4 marks

General term

First Negative Term

A pattern begins at \(100\) and decreases by \(7\) each time. Determine \(T_n\), then find the first negative term.

Memo

✓ \(a=100\), \(d=-7\), so \(T_n=100+(n-1)(-7)=107-7n\)✓ \(107-7n<0\Rightarrow n>15.28\)✓ \(T_{15}=2\) (positive), \(T_{16}=-5\) (negative) — \(\boxed{T_{16}=-5\text{ is the first negative term}}\)

15%

Level 4 | Problem Solving

Diagrams, Word Problems, and Reverse-Engineering

Read a pattern from a picture, work in context, and find \(a\) and \(d\) from two given terms.

Q14Equation Station SA Practice Question6 marks

Graph interpretation

Determine \(T_n\) From a Diagram (No Numbers Given)

The diagram shows squares joined edge to edge, built from matchsticks (no term values are written). Count the matchsticks yourself, determine \(T_n\), then find which pattern number uses \(40\) matchsticks.

Memo

✓ Counting matchsticks: \(T_1=4\), \(T_2=7\), \(T_3=10\) — constant first difference \(d=3\)✓ \(T_n=4+(n-1)(3)=3n+1\)✓ Set \(3n+1=40\): \(3n=39\Rightarrow n=\boxed{13}\)

Q15Equation Station SA Practice Question5 marks

General term

Chairs in a Hall

A hall arranges chairs in rows. Row 1 has \(8\) chairs, row 2 has \(13\), row 3 has \(18\), and row 4 has \(23\). Determine \(T_n\), then find the number of chairs in row \(20\).

Memo

✓ \(a=8\), \(d=5\), so \(T_n=8+(n-1)(5)=5n+3\)✓ \(T_{20}=100+3=\boxed{103}\) chairs

Q16Equation Station SA Practice Question7 marks

ClassificationVerification

Full Investigation

Consider the pattern \(12;\;7;\;2;\;-3;\;\ldots\) (a) Show that this is a linear pattern. (b) Determine \(T_n\). (c) Determine \(T_{40}\). (d) Is \(-500\) a term of this pattern? Justify your answer.

Memo

✓ (a) First differences: \(-5;-5;-5\) — constant, so it is linear✓ (b) \(a=12\), \(d=-5\): \(T_n=12+(n-1)(-5)=17-5n\)✓ (c) \(T_{40}=17-200=\boxed{-183}\)✓ (d) \(17-5n=-500\Rightarrow-5n=-517\Rightarrow n=103.4\); not a whole number, so \(\boxed{-500\text{ is not a term}}\)

Q17Equation Station SA Practice Question6 marks

General term

Finding \(a\) and \(d\) From Two Given Terms

The 5th term of a linear pattern is \(23\), and the 9th term is \(43\). Determine the first term and the common difference, then write down \(T_n\).

Memo

✓ \(T_5=a+4d=23\) and \(T_9=a+8d=43\)✓ Subtract: \(4d=20\Rightarrow d=5\)✓ Substitute: \(a+4(5)=23\Rightarrow a=\boxed{3}\), \(d=\boxed{5}\)✓ \(T_n=3+(n-1)(5)=\boxed{5n-2}\). Check: \(T_5=23\) ✓, \(T_9=43\) ✓