Determine This

Grade 11 Patterns Sequences — How to Identify the Question Type

Grade 11 number patterns come in two types — linear (arithmetic) and quadratic. Geometric sequences and series are Grade 12 content, not Grade 11. Real exam questions often don't tell you which type you're looking at, so before you reach for a formula, work out the type from the numbers themselves.

Type 1: Arithmetic sequences

Trigger words: "Linear pattern" (the term DBE actually uses, instead of "arithmetic sequence"), "constant difference", "Determine the general term, Tn"

Trigger structure: The difference between each term and the one before it is the same constant value, d.

Do not confuse with: Quadratic patterns — a linear pattern's first differences are constant; a quadratic pattern's first differences change, but its second differences are constant.

Method (no numbers — just the steps)

  1. Find d by subtracting consecutive terms (T2 − T1)
  2. Identify a, the first term
  3. Substitute a and d into Tn = a + (n − 1)d
  4. Simplify
Grade 11Paper 1Level 1 — Knowledge
3.1.1 Given the linear pattern: 7 ; 2 ; −3 ; ... Determine the general term, Tn, of the linear pattern.

Source: DBE Mathematics Paper 1, November 2018, Question 3.1.1

Common mistake

Calculating d as T1 − T2 instead of T2 − T1, which flips the sign of the common difference.

See the progression — same type, increasing difficulty

Easy
Given the linear pattern: 4 ; 7 ; 10 ; 13 ; ... Determine the general term, Tn, of the pattern.

Practice question — not sourced from a past paper.

Medium
3.1.2 Given the linear pattern: 7 ; 2 ; −3 ; ... Calculate the value of T20.

Source: DBE Mathematics Paper 1, November 2018, Question 3.1.2

Hard
3.2 6 ; 2x + 1 ; and 3x − 3 are the first three terms of a linear pattern. Calculate the value of x.

Source: DBE Mathematics Paper 1, November 2018, Question 3.2

Type 2: Quadratic sequences

Trigger words: "Quadratic pattern" (the term DBE uses), "second difference is constant"

Trigger structure: The first differences are NOT constant, but the differences between those first differences (the second differences) ARE constant.

Do not confuse with: Arithmetic sequences — a quadratic sequence's first differences change; only its second differences are constant.

Method (no numbers — just the steps)

  1. Calculate the first differences between consecutive terms
  2. Calculate the second differences — confirm they are constant
  3. Use 2a = (the constant second difference) to find a
  4. Use the first term and first difference to set up equations for b and c
  5. Write the general term as Tn = an² + bn + c
Grade 11Paper 1Level 3 — Complex Procedures
Given the quadratic pattern: 244 ; 193 ; 148 ; 109 ; ... 4.1 Write down the next term of the pattern. 4.2 Determine a formula for the nth term of the pattern.

Source: DBE Mathematics Paper 1, November 2017, Question 4.1–4.2

Common mistake

Stopping at the first differences and concluding the sequence is arithmetic, without checking whether the SECOND differences are the ones that are actually constant.

See the progression — same type, increasing difficulty

Easy
Given the quadratic pattern: 3 ; 7 ; 13 ; 21 ; 31 ; ... Determine the general term, Tn, of the pattern.

Practice question — not sourced from a past paper.

Medium
Given the quadratic pattern: 244 ; 193 ; 148 ; 109 ; ... Which term of the pattern will have a value of 508?

Source: DBE Mathematics Paper 1, November 2017, Question 4.3

Hard
A quadratic pattern has general term Tn = (n² + 3n + 4) / 2. If the sum of two consecutive terms in the pattern is 1227, calculate the difference between these two terms.

Source: Adapted from DBE Mathematics Paper 1, November 2018, Question 4.4

Type 3: Identify the type of pattern first

Trigger words: No label given — the question doesn't say "linear" or "quadratic"

Trigger structure: A list of numbers is given with no indication of type, often followed by "determine Tn" — you must diagnose linear vs quadratic before you can pick a formula.

Method (no numbers — just the steps)

  1. Check the first differences — if constant, it's a linear pattern
  2. If not constant, check the second differences — if THOSE are constant, it's a quadratic pattern
  3. Apply the general term formula that matches the type you found
Grade 11Paper 1Level 3 — Complex Procedures
The first four terms of a sequence are: 5 ; 8 ; 13 ; 20 Determine the general term, Tn, of the sequence.

Practice question — not sourced from a past paper.

Common mistake

Assuming the type from the 'shape' of the numbers (e.g. assuming linear just because the gaps look small and steady) instead of actually testing whether the first differences are constant.

See the progression — same type, increasing difficulty

Easy
The first four terms of a sequence are: 6 ; 10 ; 14 ; 18 Determine Tn.

Practice question — not sourced from a past paper.

Medium
The first four terms of a sequence are: 1 ; 6 ; 15 ; 28 Determine Tn.

Practice question — not sourced from a past paper.

Hard
The first four terms of a sequence are: 5 ; 7 ; 11 ; 17 ; 25 Determine Tn.

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.