Tiles that grow by two

The idea

A linear pattern is a sequence of numbers in which the difference between consecutive terms is constant; its nth term is a linear expression in n.

A growing pattern

A pattern of square tiles grows stage by stage. Each stage adds two more tiles to the stage before. Stage 1 has 1 tile, stage 2 has 3, stage 3 has 5 and stage 4 has 7.

How many tiles do you think Stage 10 will have?

• 19
• 20
• 21

Build the stages

Move from stage to stage. See the tiles grow, and read the number of tiles in the table.

The nth term

A linear pattern is a sequence of numbers in which the difference between consecutive terms is constant. Its nth term is a linear expression in n.

Each stage is two arms of n tiles that share one corner tile. So the number of tiles is one less than twice n: 2n − 1. In Stage 5 that is 2 × 5 − 1 = 9.

Notes

A linear pattern has a constant difference between consecutive terms, and its nth term is a linear expression in n (here 2n − 1).

Check yourself

How many tiles are in Stage 10 of the pattern 1, 3, 5, 7, …?

Answer: 19

2 × 10 − 1 = 20 − 1 = 19 tiles.

The nth term of 1, 3, 5, 7, … is…

Which sequence is a linear pattern?

In the linear pattern 3, 5, 7, 9, …, the nth term is 2n + 1. What is the 6th term?

Answer: 13

2 × 6 + 1 = 13. (Check: 3, 5, 7, 9, 11, 13.)

• 2n + 1. n = 1 would give 3, but the first term is 1.
• n + 2. n = 2 would give 4, but the second term is 3.
• 2n − 1 — correct. Yes. n = 1 gives 1, n = 2 gives 3, n = 3 gives 5.
• 5, 8, 11, 14, … — correct. Yes. It goes up by 3 every time.
• 1, 2, 4, 8, …. The differences are 1, 2, 4: not constant.
• 1, 4, 9, 16, …. The differences are 3, 5, 7: not constant.