Arithmetic Progressions · Answer guide

Lesson 1 · Recognise a constant difference

Distinguish an arithmetic progression from other patterns.

1. What is the common difference in 2, 5, 8, 11, …?

   Answer: 3. Subtract consecutive terms: 5 − (2) = 3. The same difference occurs again.

2. What is the common difference in 10, 8, 6, 4, …?

   Answer: -2. Subtract consecutive terms: 8 − (10) = -2. The same difference occurs again.

3. What is the common difference in -4, 1, 6, 11, …?

   Answer: 5. Subtract consecutive terms: 1 − (-4) = 5. The same difference occurs again.

Discuss: Create an increasing AP, a decreasing AP and a non-AP. Explain each using consecutive differences.

Teaching note: An AP may decrease or remain constant; d can be negative or zero. A recognisable pattern alone does not make an AP. Open responses need discussion; they are not automatically marked.

Lesson 2 · Find an AP term without listing them all

Use aₙ = a + (n − 1)d and check the index.

1. An AP has first term 3 and difference 4. Find term 8.

   Answer: 31. aₙ = a + (n − 1)d = 3 + (8 − 1) × (4) = 31.

2. An AP has first term 7 and difference 2. Find term 10.

   Answer: 25. aₙ = a + (n − 1)d = 7 + (10 − 1) × (2) = 25.

3. An AP has first term 20 and difference -2. Find term 6.

   Answer: 10. aₙ = a + (n − 1)d = 20 + (6 − 1) × (-2) = 10.

Discuss: A seating row starts with 12 chairs and each next row adds 3. Find row 20, then decide whether a row can have 61 chairs.

Teaching note: Using a + nd shifts the answer by one term. A target belongs to an AP only if the resulting n is a positive integer. Open responses need discussion; they are not automatically marked.

Lesson 3 · Add many AP terms efficiently

Derive and apply the sum formula by pairing.

1. Find the sum of the first 6 terms of the AP with a = 2, d = 2.

   Answer: 42. The last term is 12. Sₙ = n(a + last)/2 = 6 × (2 + 12)/2 = 42.

2. Find the sum of the first 8 terms of the AP with a = 3, d = 3.

   Answer: 108. The last term is 24. Sₙ = n(a + last)/2 = 8 × (3 + 24)/2 = 108.

3. Find the sum of the first 5 terms of the AP with a = 10, d = 1.

   Answer: 60. The last term is 14. Sₙ = n(a + last)/2 = 5 × (10 + 14)/2 = 60.

Discuss: Design a 7-day saving plan that is an AP. Calculate the total by listing and by the formula, and compare.

Teaching note: The last term and the number of terms are different quantities. Do not forget to halve the total after pairing two rows. Open responses need discussion; they are not automatically marked.

Final check · Set 1

Try one set after the lessons. Use the other set after reviewing any errors.

1. What is the common difference in 15, 12, 9, 6, …?

   -3. Subtract consecutive terms: 12 − (15) = -3. The same difference occurs again.

2. What is the common difference in 0, 6, 12, 18, …?

   6. Subtract consecutive terms: 6 − (0) = 6. The same difference occurs again.

3. An AP has first term -2 and difference 3. Find term 9.

   22. aₙ = a + (n − 1)d = -2 + (9 − 1) × (3) = 22.

4. An AP has first term 12 and difference 0. Find term 15.

   12. aₙ = a + (n − 1)d = 12 + (15 − 1) × (0) = 12.

5. Find the sum of the first 7 terms of the AP with a = 1, d = 4.

   91. The last term is 25. Sₙ = n(a + last)/2 = 7 × (1 + 25)/2 = 91.

6. Find the sum of the first 9 terms of the AP with a = 5, d = 0.

   45. The last term is 5. Sₙ = n(a + last)/2 = 9 × (5 + 5)/2 = 45.

Final check · Set 2

Try one set after the lessons. Use the other set after reviewing any errors.

1. What is the common difference in -9, -7, -5, -3, …?

   2. Subtract consecutive terms: -7 − (-9) = 2. The same difference occurs again.

2. What is the common difference in 20, 16, 12, 8, …?

   -4. Subtract consecutive terms: 16 − (20) = -4. The same difference occurs again.

3. An AP has first term 40 and difference -3. Find term 8.

   19. aₙ = a + (n − 1)d = 40 + (8 − 1) × (-3) = 19.

4. An AP has first term 1 and difference 6. Find term 11.

   61. aₙ = a + (n − 1)d = 1 + (11 − 1) × (6) = 61.

5. Find the sum of the first 6 terms of the AP with a = 4, d = 5.

   99. The last term is 29. Sₙ = n(a + last)/2 = 6 × (4 + 29)/2 = 99.

6. Find the sum of the first 8 terms of the AP with a = 30, d = -3.

   156. The last term is 9. Sₙ = n(a + last)/2 = 8 × (30 + 9)/2 = 156.