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Grade 12 Patterns Sequences Series — How to Identify the Question Type

Grade 12 extends Grade 11 patterns to include geometric sequences, arithmetic and geometric series, and sigma notation. The instruction word and the presence of a common ratio (r) are the fastest signals for which type you are dealing with.

Type 1: Arithmetic sequences and series

Trigger words: "Arithmetic sequence", "linear pattern", "Sn =", "sum of the first n terms"

Trigger structure: A constant first difference d is present, or the question asks for a sum.

Do not confuse with: Geometric series — if a ratio r appears instead of a difference d, the series formulas are different.

Method (no numbers — just the steps)

  1. Confirm it is arithmetic: constant first difference d = T2 − T1
  2. General term: Tn = a + (n − 1)d
  3. Sum of first n terms: Sn = n/2 × (2a + (n − 1)d) or Sn = n/2 × (a + l) where l is the last term
Grade 12Paper 1Level 2 — Routine Procedures
3.1 Given the arithmetic sequence: 5 ; 8 ; 11 ; ... 3.1.1 Determine the general term Tn. 3.1.2 Calculate the sum of the first 20 terms.

Source: DBE Mathematics Paper 1, November 2023, Question 3.1

Common mistake

Using the geometric series formula when d is constant — always identify the type first.

See the progression — same type, increasing difficulty

Easy
The arithmetic sequence is: 2 ; 5 ; 8 ; ... Find T10 and S10.

Practice question — not sourced from a past paper.

Medium
The sum of the first n terms of an arithmetic series is Sn = 3n² + n. Determine the 5th term.

Practice question — not sourced from a past paper.

Hard
The first term of an arithmetic series is 4 and the last term is 100. If the sum of all the terms is 1040, how many terms are there?

Practice question — not sourced from a past paper.

Type 2: Geometric sequences and series

Trigger words: "Geometric sequence", "common ratio", "Sn =", "r ="

Trigger structure: Each term is obtained by multiplying the previous term by a constant ratio r = T2/T1.

Do not confuse with: Arithmetic — if the differences are constant (not the ratios), use arithmetic formulas.

Method (no numbers — just the steps)

  1. Confirm it is geometric: constant ratio r = T2/T1 = T3/T2
  2. General term: Tn = ar^(n−1)
  3. Sum of first n terms: Sn = a(r^n − 1)/(r − 1) for r ≠ 1
  4. Sum to infinity (only if |r| < 1): S∞ = a/(1 − r)
Grade 12Paper 1Level 2 — Routine Procedures
3.2 The first three terms of a geometric sequence are: 4 ; 12 ; 36 ; ... Determine the sum of the first 8 terms.

Source: DBE Mathematics Paper 1, November 2022, Question 3.2

Common mistake

Applying S∞ = a/(1 − r) when |r| ≥ 1 — the series only converges when |r| < 1.

See the progression — same type, increasing difficulty

Easy
A geometric sequence has first term 2 and common ratio 3. Write down the first 4 terms and find T6.

Practice question — not sourced from a past paper.

Medium
The geometric series is: 8 + 4 + 2 + ... Calculate the sum to infinity.

Practice question — not sourced from a past paper.

Hard
The 3rd term of a geometric sequence is 20 and the 6th term is 160. Determine the common ratio and the first term.

Practice question — not sourced from a past paper.

Type 3: Sigma notation

Trigger words: "Σ" (sigma), "Write in sigma notation", "Calculate the value of"

Trigger structure: A summation symbol Σ appears, or the question asks you to express a series compactly.

Do not confuse with: A sequence — sigma notation always represents a sum (series), not individual terms.

Method (no numbers — just the steps)

  1. Identify the general term from the expression inside the sigma
  2. Determine whether the series is arithmetic or geometric
  3. Apply the appropriate sum formula with the given bounds
  4. For a partial sum starting at k > 1: calculate from k = 1 to upper bound, then subtract the sum from k = 1 to k − 1
Grade 12Paper 1Level 3 — Complex Procedures
3.3 Calculate: Σ(k=1 to 10) (3k − 1)

Source: DBE Mathematics Paper 1, November 2021, Question 3.3

See the progression — same type, increasing difficulty

Easy
Write the series 3 + 6 + 9 + 12 + ... to 15 terms in sigma notation.

Practice question — not sourced from a past paper.

Medium
Calculate: Σ(k=3 to 8) (2k + 1)

Practice question — not sourced from a past paper.

Hard
Calculate: Σ(k=1 to ∞) 3(1/2)^(k−1)

Practice question — not sourced from a past paper.

Type 4: Mixed sequence — identify the type first

Trigger words: "Determine whether ... is arithmetic or geometric", a sequence is given without a label

Trigger structure: No type is stated. You must calculate both the differences and ratios to classify before applying any formula.

Method (no numbers — just the steps)

  1. Calculate T2 − T1 and T3 − T2 — if equal, it is arithmetic
  2. Calculate T2/T1 and T3/T2 — if equal, it is geometric
  3. If neither, it is a quadratic pattern (check second differences)
  4. Only after classifying: apply the correct formula
Grade 12Paper 1Level 2 — Routine Procedures
Is the sequence 3 ; 6 ; 12 ; 24 ; ... arithmetic, geometric, or quadratic? Hence write down the 10th term.

Practice question — not sourced from a past paper.

Common mistake

Assuming a sequence is arithmetic without checking — then applying Tn = a + (n − 1)d to a geometric sequence.

See the progression — same type, increasing difficulty

Easy
State whether each sequence is arithmetic or geometric: a) 5 ; 10 ; 15 ; 20 ; ... b) 1 ; 3 ; 9 ; 27 ; ...

Practice question — not sourced from a past paper.

Medium
The sequence 2 ; x ; 8 ; ... is geometric. Find x.

Practice question — not sourced from a past paper.

Hard
The first three terms of a sequence are: (k + 2) ; (2k − 1) ; (3k − 3). Determine the value of k if the sequence is arithmetic, and then if the sequence is geometric.

Practice question — not sourced from a past paper.

Words like determine and hence appear throughout this topic — see the instruction word glossary for full definitions.