Count the groups

The idea

An AP with first term a and common difference d is a, a + d, a + 2d, …, and its nth term is tₙ = a + (n – 1) × d.

How many groups?

The squares pattern starts with 1 square. At every stage, a group of 4 squares is added: 1, 5, 9, 13, …

At stage 10, how many groups of 4 have been added to the first square?

• 9
• 10
• 11

Stages of the pattern

Move through the stages. The first square is blue; every group of 4 has its own colour.

The nth term

An AP with first term a and common difference d is a, a + d, a + 2d, …, and its nth term is tₙ = a + (n − 1) × d.

In the squares pattern a = 1 and d = 4, so tₙ = 1 + (n − 1) × 4 = 4n − 3. The first stage has no group added; stage n has n − 1 groups.

Notes

An AP with first term a and common difference d is a, a + d, a + 2d, …, and its nth term is tₙ = a + (n − 1) × d.

Check yourself

The AP 1, 4, 7, 10, … has a = 1 and d = 3. What is the 12th term?

Answer: 34

t₁₂ = 1 + (12 − 1) × 3 = 1 + 33 = 34.

An AP has a = 5 and d = 3. What is t₂₀?

Answer: 62

t₂₀ = 5 + (20 − 1) × 3 = 5 + 57 = 62.

In tₙ = a + (n − 1) × d, why is it (n − 1) and not n?

The AP 11, 7, 3, −1, −5, … has a = 11 and d = −4. What is t₆?

Answer: -9

t₆ = 11 + (6 − 1) × (−4) = 11 − 20 = −9. Check: −5 − 4 = −9.

• Because the formula starts counting at term 0. The first term is t₁. The formula counts the steps after it, and there are n − 1 of them.
• It makes no difference. It does. With n instead of n − 1, stage 1 would get one group too many.
• Because the first term a has no d added yet, so the nth term has only n − 1 steps — correct. Yes. Stage 1 has 0 groups added, stage 2 has 1, and stage n has n − 1.