Grade 12 Maths · Paper 1

Number Patterns, Sequences and Series Grade 12

Real past paper questions and answers from 2024 to 2025, with each memo split into the ticks examiners give. Try one, mark yourself, and see exactly which skill cost you marks.

past paper questions: 138 past paper questions
skills: 15 skills
videos: 15 videos

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Question 3.2 · November 2024 · Paper 1 · National (DBE) · 4 marks

A circle with radius

6

cm is drawn. A second, smaller circle is drawn through the centre of the first circle and also touches the first circle internally, as shown in the diagram. A third, smaller circle is drawn through the centre of the second circle and touches the second circle internally. The process of drawing circles continues and forms a geometric pattern. Diagram for this question

3.2 Calculate the sum of the areas of the first 10 circles.(4)

Work it out on paper first. Then mark yourself with the official memo, one tick at a time.

What's in Number Patterns, Sequences and Series Grade 12

Each skill is one thing a memo gives a tick for. Tap a skill for its own questions, notes and videos.

Sequences (Progressions)80 past paper questions

Series24 past paper questions

Sigma Notation9 past paper questions

Read and write sigma notationRead and write sums using the Σ sign, for example Σ 2n from n = 1 to 4 means 2 + 4 + 6 + 8, which has 4 terms.17 past paper questions · 3 videos

Infinite Geometric Series25 past paper questions

Applications of Sequences and Series7 past paper questions

Split a combined sequenceSeparate a sequence that mixes two patterns into its two parts, for example 1, 2, 3, 4, 5, 8 splits into 1, 3, 5 and 2, 4, 8.10 past paper questions · 4 videos

More Number Patterns, Sequences and Series Grade 12 questions and answers

Try each one on paper before you open the memo.

1. Question 2.1.2 · November 2024 · Paper 1 · National (DBE) · 4 marks
2.1 The first term of an arithmetic series is 7. The common difference of this series is 5 and the series contains 20 terms.
2.1.2 The original arithmetic series is extended to 75 terms. The sum of these 75 terms is 14 400. Using sigma notation, write down an equation for the sum of the terms added to the original series.(4)
Show the memo (3 ticks)Memo
1. ✓
  Determine general term of series $T_{n} = 7 + (n - 1) 5$
  $T_{n} = 5 n + 2$
  1 mark · Find a linear general term
2. ✓
  Calculate sum of added terms $S_{a dd e d} = S_{75} - S_{20}$
  $S_{a dd e d} = 14400 - 1090 = 13310$
  1 mark · Split a combined sequence
3. ✓
  Construct sigma notation equation $\sum_{n = 21}^{75} (5 n + 2) = 13310$
  2 marks · Read and write sigma notation
2. Question 2.1.3 · March 2024 · Paper 1 · Provincial · 4 marks
2.1 The given number pattern is a combination of a quadratic sequence and an arithmetic sequence: $16; 32; 0; 28; - 12; 24; - 20; 20; \dots$
2.1.3 The given number pattern has two consecutive terms that are equal in value. Determine the positions of the two terms.(4)
Show the memo (4 ticks)Memo
1. ✓
  Equate the two general terms $2 n^{2} - 22 n + 36 = - 4 n + 36$
  1 mark · Set up an expression or equation
2. ✓
  Simplify and factorize $2 n^{2} - 18 n = 0$
  $n^{2} - 9 n = 0$
  $n (n - 9) = 0$
  1 mark · Take out a common factor
3. ✓
  Identify valid solution for n $n = 9$ (since $n = 0$ is invalid for position)
  1 mark · Solve a quadratic by factorising
4. ✓
  Determine term positions in combined patternThe terms are in positions $2 n - 1$ and $2 n$ .
  If $n = 9$ , positions are $17$ and $18$ .
  Answer: $T_{17}$ and $T_{18}$
  1 mark · Split a combined sequence
3. Question 2.1.1 · May/June 2024 · Paper 1 · National (DBE) · 2 marks
2.1 Consider the geometric series: $4 + 2 + 1 + \frac{1}{2} + \dots$
2.1.1 Does the series converge? Justify your answer.(2)
Show the memo (2 ticks)Memo
1. ✓
  Yes, the series converges $r = \frac{1}{2}$ , so yes
  1 mark · Use the convergence condition
2. ✓
  Reason $- 1 < \frac{1}{2} < 1$
  1 mark · Use the convergence condition
4. Question 1.1.2 · March 2025 · Provincial · 2 marks
Consider the arithmetic sequence: $8; 15; 22; ...$
1.1 Consider the arithmetic sequence: $8; 15; 22; ...$
1.1.2 Calculate the sum of the first 36 terms.(2)
Show the memo (2 ticks)Memo
1. ✓
  Substitute into sum formula $S_{36} = \frac{36}{2} [2 (8) + (36 - 1) (7)]$
  1 mark · Calculate an arithmetic series sum
2. ✓
  Calculate final answer $S_{36} = 4698$
  1 mark · Calculate an arithmetic series sum
5. Question 3.1.2 · May/June 2024 · Paper 1 · National (DBE) · 4 marks
3.1 Consider the quadratic number pattern: $3; 7; 12; \dots$
3.1.2 What number must be added to $T_{n - 1}$ so that $T_{n} = 13527$ ?(4)
Show the memo (4 ticks)Memo
1. ✓
  Equate general term to target value $13527 = 0, 5 n^{2} + 2, 5 n$
  1 mark · Use a quadratic general term
2. ✓
  Solve quadratic equation for n $n = 162$
  1 mark · Use the quadratic formula
3. ✓
  Determine value of n-1 term $T_{161} = 13363$
  1 mark · Use a quadratic general term
4. ✓
  Calculate difference $13527 - 13363 = 164$
  1 mark · Use a pattern's first differences

Number Patterns, Sequences and Series Grade 12 videos

Short lessons from South African teachers first. Videos load only when you tap play, to save your data.

Area of Rectangle Given - Find the Length and Width - Solving Quadraticsmathtipxyz

Determine the rule of a numeric pattern grade 9 mathematicsALRI PRINSLOO

Gr 12 Maths - Sequences and Series - 12 - Combined SequencesJuniorTukkie at the University of Pretoria

Quadratic number patterns grade 11: introduction and examplesKevinmathscience

Quadratic Pattern Grade 11: ExamKevinmathscience

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