Grade 12 Maths · Number Patterns, Sequences and Series

Calculate the sum to infinity

Find the total of a geometric series that goes on forever using S = a/(1 − r), for example 8 + 4 + 2 + … = 16.

past paper questions use it: 10 past paper questions use it
videos: 3 videos

Learn it: calculate the sum to infinity

Pick one and watch it, then try the questions below. Videos load only when you tap play.

12) Sum to Infinity Grade 12 | IntroKevinmathscience

12f) Sum to Infinity Grade 12 | TryKevinmathscience

Patterns and Sequences: Lesson: Sum of an Infinite SeriesMindset

Now try it on a real exam question

No sign-up. Takes 3 minutes.

Question 3.2 · March 2023 · Provincial · 2 marks

A convergent geometric series is given by:

\frac{5 ( x + 1 )}{3} + \frac{5 ( x + 1 ) ^{2}}{9} + \frac{5 ( x + 1 ) ^{3}}{27} + ...

3.2 If

x = 1

, calculate the sum to infinity,

S_{\infty}

.(2)

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

More past paper questions: calculate the sum to infinity

Work each one out first, then open the memo and check every tick.

3. Question 2.2.3 · May/June 2025 · Paper 1 · National (DBE) · 3 marks
Given the arithmetic series: $5 + 7 + 9 + ... + 93$
2.2 The general term of a geometric sequence is $T_{n} = 2^{n + 2}$
2.2.3 Calculate $\sum_{n = 1}^{\infty} \frac{1}{T _{n}}$ (3)
Show the memo (3 ticks)Memo
1. ✓
  Determine the seriesThe series is: $\frac{1}{8} + \frac{1}{16} + \frac{1}{32} + ...$
  1 mark · Find a geometric sequence term
2. ✓
  Calculate the sum to infinity $S_{\infty} = \frac{a}{1 - r}$
  $= \frac{\frac{1}{8}}{1 - \frac{1}{2}}$
  1 mark · Calculate the sum to infinity
3. ✓
  Final result $S_{\infty} = \frac{1}{4}$
  1 mark · Calculate the sum to infinity
4. 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
5. Question 1.2.2 · March 2022 · Provincial · 3 marks
1.2 Given: $\sum_{k = 1}^{\infty} 4 (0, 2)^{k - 1}$
1.2.2 Calculate sum to infinity of the series.(3)
Show the memo (3 ticks)Memo
1. ✓
  Identify parameters $a = 4$ , $r = 0, 2$ (or $\frac{1}{5}$ )
  1 mark · Read and write sigma notation
2. ✓
  Apply formula $S_{\infty} = \frac{a}{1 - r} = \frac{4}{1 - 0 , 2}$
  1 mark · Calculate the sum to infinity
3. ✓
  Calculate sum $S_{\infty} = 5$
  1 mark · Calculate the sum to infinity
6. Question 3.1.2 · September 2024 · Paper 1 · Provincial · 4 marks
3.1 Given the geometric sequence: $8 (x - 2)^{2}; 4 (x - 2)^{3}; 2 (x - 2)^{4}; ...$ $x \neq = 2$
3.1.2 Determine the sum to infinity of the series if $x = 2, 5$ .(4)
Show the memo (4 ticks)Memo
1. ✓
  Value of a $a = 2$
  1 mark · Substitute into a formula
2. ✓
  Value of r $r = \frac{1}{4}$
  1 mark · Find the common ratio
3. ✓
  Substitute into Sum formula $S_{\infty} = \frac{2}{1 - 0 , 25}$
  1 mark · Calculate the sum to infinity
4. ✓
  Answer $\frac{8}{3}$
  1 mark · Calculate the sum to infinity

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