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

Infinite Geometric Series Grade 12

Real Infinite Geometric Series past paper questions and answers from 2022 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: 25 past paper questions
skills: 3 skills
videos: 3 videos

Try a Infinite Geometric Series exam question

No sign-up. Takes 3 minutes.

Question 2.2 · March 2023 · Provincial · 3 marks

2.2 Calculate the value of:

\sum_{r = 2}^{\infty} 3 \cdot 2^{1 - r} + \sum_{r = 2}^{12} 3 \cdot 2^{1 - r}

(give your answer to 3 decimal places).(3)

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

Infinite Geometric Series skills

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

More Infinite Geometric Series Grade 12 questions and answers

Try each one on paper before you open the memo.

4. 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
5. Question 1.2.2 · March 2025 · Provincial · 2 marks
1.2 $(x - 2)^{2} + (x - 2)^{3} + (x - 2)^{4} + ...$ forms a geometric series.
1.2.2 Determine the value(s) of $x$ for which the series will converge.(2)
Show the memo (2 ticks)Memo
1. ✓
  Apply convergence condition $- 1 < r < 1$
  $- 1 < x - 2 < 1$
  1 mark · Use the convergence condition
2. ✓
  Solve for x $1 < x < 3$
  1 mark · Solve a linear inequality
6. Question 2.1.3 · March 2025 · Provincial · 2 marks
2.1 Given: $\sum_{k = 2}^{13} (- 3)^{k}$
2.1.3 Will $\sum_{k = 2}^{\infty} (- 3)^{k}$ converge? Explain your answer.(2)
Show the memo (2 ticks)Memo
1. ✓
  State convergence statusNo, the series will not converge.
  1 mark · Use the convergence condition
2. ✓
  Provide reason $r = - 3$ , which is not between -1 and 1 ( $r < - 1$ or $∣ r ∣ \geq 1$ ).
  1 mark · Use the convergence condition
7. Question 1.2.3 · March 2024 · Paper 1 · Provincial · 3 marks
1.2 A frog is making a series of jumps. With every next jump, he has only enough energy left to jump $\frac{2}{3}$ the distance of his previous jump.
1.2.3 If the frog continues to jump in this way, will he be able to catch a trapped insect that is $230$ cm away from his starting point? Show all your calculations.(3)
Show the memo (3 ticks)Memo
1. ✓
  Calculate sum to infinity $S_{\infty} = \frac{a}{1 - r}$
  $S_{\infty} = \frac{81}{1 - \frac{2}{3}}$
  1 mark · Calculate the sum to infinity
2. ✓
  Determine the limit of the sum $S_{\infty} = 243$ cm
  1 mark · Calculate the sum to infinity
3. ✓
  Conclusion based on comparison $243$ cm $> 230$ cm
  Yes, he will be able to catch the insect.
  1 mark · Calculate the sum to infinity
8. Question 3.1.1 · September 2024 · Paper 1 · Provincial · 3 marks
3.1 Given the geometric sequence: $8 (x - 2)^{2}; 4 (x - 2)^{3}; 2 (x - 2)^{4}; ...$ $x \neq = 2$
3.1.1 Determine the value(s) of x where the sequence converges.(3)
Show the memo (3 ticks)Memo
1. ✓
  Expression for r $r = \frac{x - 2}{2}$
  1 mark · Find the common ratio
2. ✓
  Convergence inequality $- 1 < \frac{x - 2}{2} < 1$
  1 mark · Use the convergence condition
3. ✓
  Answer $0 < x < 4$
  1 mark · Solve a linear inequality
9. Question 3.1.2 · March 2023 · Provincial · 2 marks
3.1 $(x - 2)^{2} + (x - 2)^{3} + (x - 2)^{4} + ...$ form a geometric series.

Geometric series properties
3.1.2 Determine the value(s) of $x$ for which the series will converge.(2)
Show the memo (2 ticks)Memo
1. ✓
  Apply convergence condition $- 1 < r < 1$
  $- 1 < x - 2 < 1$
  1 mark · Use the convergence condition
2. ✓
  Solve for x $1 < x < 3$
  1 mark · Solve a linear inequality
10. Question 3.1.2 · March 2022 · Provincial · 4 marks
3.1 Given: $\sum_{k = 1}^{\infty} (\frac{1}{p})^{k - 1}$
3.1.2 Determine the values of $p$ for which the series is converging.(4)
Show the memo (3 ticks)Memo
1. ✓
  State condition for convergence $- 1 < r < 1$
  1 mark · Use the convergence condition
2. ✓
  Substitute r $- 1 < \frac{1}{p} < 1$
  1 mark · Use the convergence condition
3. ✓
  Solve inequality $p < - 1$ or $p > 1$
  2 marks · Solve a linear inequality

Infinite Geometric Series Grade 12 videos

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

4)Gr 12 Geometric Sequence | IntroKevinmathscience

11) Converging series Gr 12 | Intro Part 1Kevinmathscience

12) Sum to Infinity Grade 12 | IntroKevinmathscience

Free on Android

Know exactly which skill is costing you marks.

Mark past papers tick by tick in the NavyBlue app. It remembers every mark you lose, shows you the skills to fix first, and picks your next questions.

The NavyBlue app showing a learner's weak skills