Grade 12 Maths · Number Patterns, Sequences and Series

Calculate a geometric series sum

Add the terms of a pattern with a constant ratio using Sn = a(rⁿ − 1)/(r − 1), for example 1 + 2 + 4 + 8 = 15.

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8) Sum of Geometric Series Gr 12 | IntroKevinmathscience

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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.

More past paper questions: calculate a geometric series sum

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3. Question 2.2.4 · May/June 2025 · Paper 1 · National (DBE) · 4 marks
Given the arithmetic series: $5 + 7 + 9 + ... + 93$
2.2 The general term of a geometric sequence is $T_{n} = 2^{n + 2}$
2.2.4 Consider the first $21$ terms of the sequence $T_{n} = 2^{n + 2}$ . Calculate the sum of the terms in this sequence that are not powers of $4$ .(4)
Show the memo (3 ticks)Memo
1. ✓
  Identify series properties and calculate sum of all 21 termsTotal sequence: $8 + 16 + 32 + ...$ (up to 21 terms)
  Powers of 4 subset: $16 + 64 + 256 + ...$ (up to 10 terms within the 21)
  Total terms $S_{21} = \frac{8 ( 2 ^{21} - 1 )}{2 - 1}$
  1 mark · Calculate a geometric series sum
2. ✓
  Calculate sum of terms that are powers of 4Sum of powers of 4: $S_{10} = \frac{16 ( 4 ^{10} - 1 )}{4 - 1}$
  2 marks · Calculate a geometric series sum
3. ✓
  Subtract sum of powers of 4 from total sum $S_{diff} = S_{21} - S_{10}$
  $= 16777208 - 5592400$
  $= 11184808$
  1 mark · Split a combined sequence
4. Question 2.2 · May/June 2024 · Paper 1 · National (DBE) · 5 marks
2.2 Given: $\sum_{p = k}^{10} 3^{p - 1} = 29520$ . Calculate the value of $k$ .(5)
Show the memo (5 ticks)Memo
1. ✓
  Write out the first terms: $3^{k - 1} + 3^{k}$ $\sum_{p = k}^{10} 3^{p - 1} = 3^{k - 1} + 3^{k + 1 - 1} + 3^{k + 2 - 1} + \dots + 3^{9}$
  $= 3^{k - 1} + 3^{k} + 3^{k + 1} + \dots + 3^{9}$
  1 mark · Read and write sigma notation
2. ✓
  Common ratio $r = 3$
  1 mark · Read and write sigma notation
3. ✓
  Substitution of $3^{k - 1}$ into the correct sum formula $S_{n} = \frac{a ( r ^{n} - 1 )}{r - 1}$
  $29520 = \frac{3 ^{k - 1} ( 3 ^{11 - k} - 1 )}{3 - 1}$
  1 mark · Calculate a geometric series sum
4. ✓
  Number of terms $n = 10 - k + 1 = 11 - k$
  1 mark · Read and write sigma notation
5. ✓
  Answer $3^{10} - 3^{k - 1} = 59040$
  $3^{k - 1} = 9$
  $k - 1 = 2$
  $k = 3$
  1 mark · Solve an exponential equation
5. Question 2.1.4 · March 2025 · Provincial · 3 marks
2.1 Given: $\sum_{k = 2}^{13} (- 3)^{k}$
2.1.4 Calculate $\sum_{k = 2}^{13} (- 3)^{k} x$ . Give your answer in terms of $x$ .(3)
Show the memo (3 ticks)Memo
1. ✓
  Identify series parametersThe series is $9 x - 27 x + 81 x ...$
  $a = 9 x$
  $r = - 3$
  $n = 13 - 2 + 1 = 12$
  1 mark · Read and write sigma notation
2. ✓
  Substitute into geometric sum formula $S_{12} = \frac{9 x (( - 3 ) ^{12} - 1 )}{- 3 - 1}$
  1 mark · Calculate a geometric series sum
3. ✓
  Calculate final answer $S_{12} = - 1195740 x$
  1 mark · Calculate a geometric series sum
6. Question 3.2.2 · September 2024 · Paper 1 · Provincial · 3 marks
3.2 Given: $\sum_{k = 3}^{12} 3 (- 2)^{k - 2}$
3.2.2 Calculate the sum of the series.(3)
Show the memo (3 ticks)Memo
1. ✓
  Determine a and r $a = - 6; r = - 2$
  1 mark · Read and write sigma notation
2. ✓
  Substitute into Sn formula $S_{10} = \frac{- 6 ( 1 - ( - 2 ) ^{10} )}{1 - ( - 2 )}$
  1 mark · Calculate a geometric series sum
3. ✓
  Answer $2046$
  1 mark · Calculate a geometric series sum

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