Grade 10 Maths · Number Patterns, Sequences and Series

Split a combined sequence

Separate a sequence that mixes two patterns into its two parts, for example 1, 2, 3, 4, 5, 8 splits into 1, 3, 5 and 2, 4, 8.

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Gr 12 Maths - Sequences and Series - 12 - Combined SequencesJuniorTukkie at the University of Pretoria

Combined Number Patterns (Grade 12 Math)MatheNatic

16) Combined Arithmetic Geometric | IntroKevinmathscience

Combined sequences —odd-Arithmetic terms & even-Geometric terms (TT test question 6)AlphaMath'

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Question 1.3 · March 2022 · Provincial · 5 marks

1.3 Evaluate the sum of:

3 + 11 + 3 + 15 + 3 + 19 + \dots + 107

(5)

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More past paper questions: split a combined sequence

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3. 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
4. Question 1.2.2 · March 2025 · Provincial · 4 marks
1.2 Consider the sequence: $7; 7; 7; 12; 7; 17; 7; 22; ...$
1.2.2 Calculate the sum of the first 103 terms of this sequence.(4)
Show the memo (4 ticks)Memo
1. ✓
  Determine number of odd and even terms103 terms consist of 52 odd terms and 51 even terms.
  1 mark · Split a combined sequence
2. ✓
  Sum the odd terms (constant 7) $S u m_{o dd} = 52 \times 7 = 364$
  1 mark · Split a combined sequence
3. ✓
  Sum the even terms (arithmetic) $S u m_{e v e n} = \frac{51}{2} [2 (7) + (51 - 1) (5)] = 6732$
  1 mark · Calculate an arithmetic series sum
4. ✓
  Calculate total sum $T o t a lS u m = 364 + 6732 = 7096$
  1 mark · Split a combined sequence
5. Question 2.1.3 · March 2025 · Provincial · 4 marks
The given number pattern is a combination of a quadratic sequence and an arithmetic sequence: $16; 32; 0; 28; - 12; 24; - 20; 20; ...$
2.1 Pattern analysis questions.
2.1.3 The given number pattern has two consecutive terms that are equal in value, determine the positions of the two terms.(4)
Show the memo (4 ticks)Memo
1. ✓
  Equate the two general terms $2 n^{2} - 22 n + 36 = - 4 n + 36$
  1 mark · Split a combined sequence
2. ✓
  Simplify to standard form $2 n^{2} - 18 n = 0$
  1 mark · Write a quadratic in standard form
3. ✓
  Factorize $2 n (n - 9) = 0$
  1 mark · Take out a common factor
4. ✓
  Determine positions $n = 9$ (since $n \neq = 0$ )
  The terms are at positions $T_{17}$ and $T_{18}$ (Odd position $2 n - 1 = 17$ , Even position $2 n = 18$ for $n = 9$ )
  1 mark · Split a combined sequence
6. Question 2.1.3 · March 2023 · Provincial · 3 marks
$10; a; 24; b; 38; ...$ are the first five terms of an arithmetic progression.
2.1 Arithmetic progression questions
2.1.3 If there are 67 terms in this arithmetic sequence, determine the sum of all the even terms of this sequence.(3)
Show the memo (3 ticks)Memo
1. ✓
  Sequence of the even terms $10; 24; 38; 52; 66; ...$
  $a = 10$ ; $d = 14$ ; $n = 34$
  1 mark · Split a combined sequence
2. ✓
  Substitute into the sum formula $S_{n} = \frac{n}{2} [2 a + (n - 1) d]$
  $S_{34} = \frac{34}{2} [2 (10) + (34 - 1) (14)]$
  1 mark · Calculate an arithmetic series sum
3. ✓
  Answer $S_{34} = 8194$
  1 mark · Calculate an arithmetic series sum

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