Grade 10 Number Patterns (Linear Sequences)

Grade 10 number patterns: finding the general term Tₙ = bn + c of a linear sequence, finding a term and the position of a term, with worked examples.

Grade 10 number patterns are linear sequences: patterns with a constant difference between consecutive terms. You learn to write the general term Tₙ = bn + c, use it to find any term, and work backwards to find which position a given value is in. The section is worth about 15 marks of Paper 1.

This is the start of a three-year thread. Grade 11 moves on to quadratic patterns with a constant second difference, and Grade 12 adds arithmetic and geometric series with sigma notation. A clear Grade 10 method makes both of those much easier.

Key concepts

  • Linear pattern: the first difference is constant
  • General term Tₙ = bn + c, where b is the constant difference
  • Finding c by substituting n = 1 and T₁
  • Finding the position n of a given term by solving Tₙ = value
  • Patterns in pictures and tables, e.g. matchsticks and tiles

Worked examples

Find the general term of 2; 7; 12; 17; …

The constant difference is 5, so Tₙ = 5n + c. T₁ = 2 gives 5 + c = 2, so c = −3 and Tₙ = 5n − 3.

Using Tₙ = 5n − 3, find T₁₅.

Substitute n = 15: T₁₅ = 5(15) − 3 = 75 − 3 = 72.

Which term of 4; 10; 16; … equals 100?

The difference is 6 and T₁ = 4, so Tₙ = 6n − 2. Solve 6n − 2 = 100: 6n = 102 and n = 17. It is the 17th term.

Common mistakes

  • Using the first term as c, writing Tₙ = 5n + 2 for 2; 7; 12; …
  • Giving a non-whole-number position; n must be a natural number.
  • Confusing the term number n with the term value Tₙ.
  • Not checking the formula against the first two terms.

Exam tips

  • Always test your formula with n = 1 and n = 2 before you use it.
  • If solving for n gives a fraction, the value is not a term of the pattern; say so.
  • For picture patterns, draw a table of n and the count before looking for a rule.

Frequently asked questions

What is the difference between a linear and a quadratic pattern?

A linear pattern has a constant first difference (Grade 10). A quadratic pattern has a constant second difference, meaning the differences themselves form a linear pattern (Grade 11).

Is 'number patterns' the same as sequences and series?

Grade 10 and 11 call it number patterns. In Grade 12 the topic becomes sequences and series, where you also add up terms.

How many marks are number patterns in Grade 10?

About 15 marks of the 100-mark Paper 1, according to the CAPS examination guidelines.

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