Grade 11 Number Patterns (Quadratic Sequences)

Key concepts

• Quadratic pattern: constant second difference
• 2a = second difference
• 3a + b = first of the first differences
• a + b + c = T₁
• Finding the position n by solving Tₙ = value (n must be a natural number)
• The first differences of a quadratic pattern form a linear pattern

Worked examples

Find the next term of 2; 5; 10; 17; …

First differences: 3; 5; 7. Second difference: 2. The next first difference is 9, so the next term is 17 + 9 = 26.

Find the general term of 2; 5; 10; 17; …

2a = 2, so a = 1. 3a + b = 3, so b = 0. a + b + c = 2, so c = 1. Tₙ = n² + 1. Check: T₃ = 9 + 1 = 10 ✓.

Tₙ = n² + 3n − 1. Find T₁₀.

T₁₀ = 10² + 3(10) − 1 = 100 + 30 − 1 = 129.

Which term of Tₙ = n² + 1 equals 401?

Solve n² + 1 = 401: n² = 400, so n = 20 (reject n = −20, since a position can't be negative). It is the 20th term.

Common mistakes

• Using the second difference as a instead of 2a = second difference.
• Keeping a negative or fractional value of n as a term position.
• Using the wrong first difference (it must be T₂ − T₁) in 3a + b.
• Not checking the formula against the given terms.

Exam tips

• Set out the terms, first differences and second differences in three neat rows.
• Check your Tₙ with n = 1, 2 and 3 before answering the rest of the question.
• If solving for n gives a non-natural number, say clearly that the value is not a term.

Frequently asked questions

How do I know if a pattern is quadratic?

Calculate the first differences. If they aren't constant, calculate the differences between them. A constant second difference means the pattern is quadratic.

Is there a quick formula for a, b and c?

Yes: 2a equals the second difference, 3a + b equals the first first-difference (T₂ − T₁), and a + b + c equals T₁. Solve them in that order.

How many marks are number patterns in Grade 11?

About 25 of the 150 marks in Paper 1, according to the CAPS examination guidelines.