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

Sigma Notation Grade 12

Real Sigma Notation past paper questions and answers from 2021 to 2024, with each memo split into the ticks examiners give. Try one, mark yourself, and see exactly which skill cost you marks.

past paper questions: 9 past paper questions
skill: 1 skill
videos: 1 videos

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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.

Sigma Notation skills

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

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

More Sigma Notation Grade 12 questions and answers

Try each one on paper before you open the memo.

4. 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
5. Question 1.3 · March 2022 · Provincial · 5 marks
1.3 Evaluate the sum of: $3 + 11 + 3 + 15 + 3 + 19 + \dots + 107$ (5)
Show the memo (5 ticks)Memo
1. ✓
  Separate the sequencesSequence 1 (Odd positions): $3; 3; 3; \dots$
  Sequence 2 (Even positions): $11; 15; 19; \dots; 107$
  1 mark · Split a combined sequence
2. ✓
  Find number of terms in AP $T_{n} = 4 n + 7$
  $107 = 4 n + 7$
  $100 = 4 n$
  $n = 25$
  1 mark · Find a linear general term
3. ✓
  Calculate sum of AP $S_{25} = \frac{25}{2} (11 + 107) = \frac{25}{2} (118) = 1475$
  1 mark · Calculate an arithmetic series sum
4. ✓
  Calculate sum of constant sequenceThere are 25 terms of 3s.
  Sum = $3 \times 25 = 75$
  1 mark · Split a combined sequence
5. ✓
  Total Sum $1475 + 75 = 1550$
  1 mark · Split a combined sequence
6. Question 1.3 · March 2021 · Provincial · 5 marks
1.3 Evaluate: $\sum_{k = 1}^{\infty} (\frac{1}{3})^{k} - \sum_{k = 2}^{40} 5$ (5)
Show the memo (3 ticks)Memo
1. ✓
  Calculate sum to infinity for first part $\sum_{k = 1}^{\infty} (\frac{1}{3})^{k} = \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + ...$
  $S_{\infty} = \frac{\frac{1}{3}}{1 - \frac{1}{3}} = \frac{1}{2}$
  2 marks · Calculate the sum to infinity
2. ✓
  Calculate sum of constant series $\sum_{k = 2}^{40} 5 = 5 + 5 + ... + 5$
  $n = 40 - 2 + 1 = 39$
  $S_{39} = 5 (39) = 195$
  1 mark · Read and write sigma notation
3. ✓
  Subtract sums $S_{\infty} - S_{39} = \frac{1}{2} - 195 = - 194, 5$ or $\frac{- 389}{2}$
  2 marks · Split a combined sequence
7. Question 3.2 · March 2021 · Provincial · 4 marks
3.2 Calculate: $\sum_{n = 3}^{\infty} 5 (3)^{1 - n}$ (4)
Show the memo (4 ticks)Memo
1. ✓
  Determine first term $n = 3 : T_{1} = 5 (3)^{1 - 3} = 5 (3)^{- 2} = \frac{5}{9}$
  1 mark · Read and write sigma notation
2. ✓
  Determine second term and ratio $n = 4 : T_{2} = 5 (3)^{1 - 4} = 5 (3)^{- 3} = \frac{5}{27}$
  $r = \frac{1}{3}$
  1 mark · Find the common ratio
3. ✓
  Apply sum to infinity formula $S_{\infty} = \frac{a}{1 - r} = \frac{\frac{5}{9}}{1 - \frac{1}{3}}$
  1 mark · Calculate the sum to infinity
4. ✓
  Final answer $= \frac{5}{6}$ (or $0, 83$ )
  1 mark · Calculate the sum to infinity
8. Question 3.2.1 · September 2024 · Paper 1 · Provincial · 1 mark
3.2 Given: $\sum_{k = 3}^{12} 3 (- 2)^{k - 2}$
3.2.1 How many terms are there in this series?(1)
Show the memo (1 tick)Memo
1. ✓
  Answer $n = 10$
  1 mark · Read and write sigma notation
9. Question 2.3 · March 2021 · Provincial · 1 mark
Given the arithmetic series: $3 + 10 + 17 + \dots + 150$ .
2.3 Express the series in sigma notation.(1)
Show the memo (1 tick)Memo
1. ✓
  Find n and write notation $150 = 7 n - 4 ⟹ 154 = 7 n ⟹ n = 22$
  $\sum_{k = 1}^{22} (7 k - 4)$
  1 mark · Read and write sigma notation

Sigma Notation Grade 12 videos

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14) Sigma Grade 12 | IntroKevinmathscience

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