Grade 12 Maths · Number Patterns, Sequences and Series · Paper 1

Arithmetic and Geometric Sequences Grade 12

Real Arithmetic and Geometric Sequences 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: 80 past paper questions
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videos: 9 videos

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Question 3.1 · May/June 2025 · Paper 1 · National (DBE) · 3 marks

Given the quadratic sequence:

14; 9; 6; 5; ...

3.1 Show that the general term of this sequence is

T_{n} = n^{2} - 8 n + 21

.(3)

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

Arithmetic and Geometric Sequences skills

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More Arithmetic and Geometric Sequences Grade 12 questions and answers

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4. Question 2.2.1 · November 2024 · Paper 1 · National (DBE) · 3 marks
2.2 The sequence of the first differences of a quadratic pattern is: $1; 3; 5; ...$
2.2.1 If $T_{99}$ of this quadratic pattern is $9632$ , calculate the value of $T_{98}$ .(3)
Show the memo (3 ticks)Memo
1. ✓
  Calculate the 98th first differenceFirst differences form arithmetic sequence: $T_{n} = 2 n - 1$
  $9 8^{t h}$ difference $= 2 (98) - 1 = 195$
  1 mark · Use a pattern's first differences
2. ✓
  Subtract difference from T99 $T_{99} - T_{98} = 1^{s t} difference_{98}$
  $T_{98} = 9632 - 195$
  1 mark · Use a pattern's first differences
3. ✓
  Final Answer $T_{98} = 9437$
  1 mark · Use a pattern's first differences
5. Question 3.1.1 · May/June 2024 · Paper 1 · National (DBE) · 3 marks
3.1 Consider the quadratic number pattern: $3; 7; 12; \dots$
3.1.1 Show that the general term of this number pattern is given by $T_{n} = \frac{1}{2} n^{2} + \frac{5}{2} n$ .(3)
Show the memo (3 ticks)Memo
1. ✓
  Determine second difference $2 a = 1 ⟹ a = 0, 5$
  1 mark · Find a quadratic general term
2. ✓
  Solve for b using first difference $3 a + b = 4 ⟹ b = 2, 5$
  1 mark · Find a quadratic general term
3. ✓
  Solve for c using first term $a + b + c = 3 ⟹ c = 0$
  1 mark · Find a quadratic general term
6. Question 1.1.1 · March 2025 · Provincial · 2 marks
1.1 Consider the arithmetic sequence: $8; 15; 22; ...$

Consider the arithmetic sequence: $8; 15; 22; ...$
1.1.1 Determine the $3 6^{th}$ term.(2)
Show the memo (2 ticks)Memo
1. ✓
  Identify parameters and substitute into formula $a = 8$ ; $d = 7$ ; $n = 36$
  $T_{36} = 8 + (36 - 1) (7)$
  1 mark · Find a linear general term
2. ✓
  Calculate final answer $T_{36} = 253$
  1 mark · Find a linear general term
7. Question 2.1.1 · March 2025 · Provincial · 4 marks
2.1 The given number pattern is a combination of a quadratic sequence and an arithmetic sequence: $16; 32; 0; 28; - 12; 24; - 20; 20; ...$

Pattern analysis questions.
2.1.1 Determine the general term of the quadratic sequence.(4)
Show the memo (5 ticks)Memo
1. ✓
  Extract quadratic sequence and find differencesSequence: $16; 0; - 12; - 20$
  First differences: $- 16; - 12; - 8$
  Second difference: $4$
  0 marks · Find a quadratic general term
2. ✓
  Calculate coefficient a $2 a = 4$
  $a = 2$
  1 mark · Find a quadratic general term
3. ✓
  Calculate coefficient b $3 a + b = - 16$
  $3 (2) + b = - 16$
  $b = - 22$
  1 mark · Find a quadratic general term
4. ✓
  Calculate coefficient c $a + b + c = 16$
  $2 - 22 + c = 16$
  $c = 36$
  1 mark · Find a quadratic general term
5. ✓
  Write final general term $T_{n} = 2 n^{2} - 22 n + 36$
  1 mark · Find a quadratic general term
8. Question 1.1.1 · March 2025 · Provincial · 2 marks
1.1 Given: The sum to $n$ terms of an arithmetic sequence is $S_{n} = 3 n^{2} - 5 n$ .
1.1.1 Calculate the sum of the first 21 terms of this sequence.(2)
Show the memo (2 ticks)Memo
1. ✓
  Substitute $n = 21$ into the formula $S_{21} = 3 (21)^{2} - 5 (21)$
  1 mark · Substitute into a formula
2. ✓
  Calculate final answer $S_{21} = 1218$
  1 mark · Substitute into a formula
9. Question 2.1.1 · March 2025 · Provincial · 2 marks
2.1 Given: $\sum_{k = 2}^{13} (- 3)^{k}$
2.1.1 Write down the values of the first three terms of the series.(2)
Show the memo (2 ticks)Memo
1. ✓
  Calculate first term (k=2) $(- 3)^{2} = 9$
  1 mark · Read and write sigma notation
2. ✓
  Calculate next two terms (k=3, 4) $(- 3)^{3} = - 27$ ; $(- 3)^{4} = 81$
  1 mark · Read and write sigma notation
10. Question 1.1.1 · March 2024 · Paper 1 · Provincial · 2 marks
1.1 Consider the arithmetic sequence: $8; 15; 22; \dots$
1.1.1 Determine the $3 6^{th}$ term(2)
Show the memo (2 ticks)Memo
1. ✓
  Identify parameters and substitute into arithmetic general term formula $a = 8$
  $d = 7$
  $n = 36$
  $T_{n} = a + (n - 1) d$
  $T_{36} = 8 + (36 - 1) (7)$
  1 mark · Find a linear general term
2. ✓
  Calculate final answer $T_{36} = 253$
  1 mark · Find a linear general term

Arithmetic and Geometric Sequences 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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