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Linear patterns · 2 marks

Observe the following growing pattern of square tiles.
Predict the number of squares in the next three stages of the pattern and write the sequence of numbers up to Stage 7 of the pattern.

Stage 1Stage 2Stage 3Stage 4
Answer: Stages 5, 6 and 7 have 9, 11 and 13 squares. The sequence up to Stage 7 is 1, 3, 5, 7, 9, 11, 13.

Step-by-step solution

Idea: Count the tiles in the stages shown (1, 3, 5, 7) and look at how each stage grows: every new stage adds one tile on top of each column, that is, 2 more tiles.

Stage 59 tilesStage 611 tilesStage 713 tiles
  1. Count the tiles shown: Stage 1 → 1, Stage 2 → 3, Stage 3 → 5, Stage 4 → 7.½ mark
  2. Each stage adds one tile on top of each of the two columns: 2 more tiles every time.½ mark
  3. So Stage 5 → 7 + 2 = 9, Stage 6 → 9 + 2 = 11, Stage 7 → 11 + 2 = 13.½ mark
  4. Sequence up to Stage 7: 1, 3, 5, 7, 9, 11, 13 (the odd numbers).½ mark
Stage 5: 9 tiles, Stage 6: 11 tiles, Stage 7: 13 tiles. Sequence: 1, 3, 5, 7, 9, 11, 13.

Check: Stage n has a left column of n − 1 tiles and a right column of n tiles, so (n − 1) + n = 2n − 1 tiles. For n = 7: 13 ✓.

Answer to write in the exam

Stages 1 to 4: 1, 3, 5, 7 tiles (each stage adds 2 tiles)

Stage 5 = 7 + 2 = 9; Stage 6 = 9 + 2 = 11; Stage 7 = 11 + 2 = 13

∴ Sequence: 1, 3, 5, 7, 9, 11, 13

Common mistakes that cost marks

  • Adding 1 tile per stage (giving 8, 9, 10). Look again: both columns grow, so 2 tiles are added.
  • Doubling each stage (7, 14, 28). The pattern adds the same number each time; it does not multiply.
  • Writing only the three new numbers when the question asks for the whole sequence up to Stage 7.

How this can come in the exam

MCQ (1 mark)

A pattern of matchstick squares in a row uses 4, 7, 10, 13, … matchsticks. The 6th term is

  1. 16
  2. 18
  3. 19
  4. 22
Show answer

(C) 19
It grows by 3 each time: 13, 16, 19. The 6th term is 19.

Short answer (2 marks)

A pattern of dots has 2, 6, 10, 14 dots in its first four stages. Find the number of dots in stages 5, 6 and 7.

Show answerThe difference is 4 each time (1 mark). Stages 5, 6, 7: 18, 22, 26 (1 mark).

Try one yourself

A tile pattern has 5, 8, 11, 14 tiles in stages 1 to 4. Write the sequence up to stage 7.

Show answer

It grows by 3: 5, 8, 11, 14, 17, 20, 23.

More questions like this

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← If the length of a rectangle is three more than twice its width and itThis leads us to conclude that the number of squares at Stage n is giv →