Arithmetic Progressions · Learner pages

Lesson 1 · Recognise a constant difference

Distinguish an arithmetic progression from other patterns.

Name or initials: ____________   Date: ____________

A sequence is an ordered list; the order matters.

1. Compare neighbours. For 5, 9, 13, 17, subtract 5 from 9 to get 4.
2. Repeat the difference check. 13 − 9 = 4 and 17 − 13 = 4, so the difference is constant.
3. Continue in the same way. Add 4 to 17 to get 21. Moving backward from 5 subtracts 4 to give 1.
4. Check a non-example. 2, 4, 8, 16 has differences 2, 4, 8. It is not an AP even though it follows a pattern.

Worked example: For 18, 13, 8, 3, the differences are all −5, so a = 18 and d = −5. The next term is −2.

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

   1. 3 · 2. 4 · 3. 2

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

   1. -1 · 2. -3 · 3. -2

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

   1. 4 · 2. 5 · 3. 6

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

Lesson 2 · Find an AP term without listing them all

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

Name or initials: ____________   Date: ____________

Going from term 1 to term 2 uses one jump; term 1 to term 5 uses four jumps.

1. Identify the start and jump. A row pattern begins 6, 10, 14, …, so a = 6 and d = 4.
2. Count the jumps. The eighth term is seven jumps from the first, not eight.
3. Calculate the term. a₈ = 6 + 7 × 4 = 34.
4. Reverse-check a target. To locate 50, solve 6 + (n − 1)4 = 50. This gives n − 1 = 11, so n = 12.

Worked example: In 9, 14, 19, …, term 15 is 9 + 14 × 5 = 79. To test 81, solve n − 1 = (81 − 9)/5 = 14.4; since n is not an integer, 81 is not a term.

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

   1. 31 · 2. 35 · 3. 27

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

   1. 27 · 2. 23 · 3. 25

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

   1. 12 · 2. 10 · 3. 8

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.

Lesson 3 · Add many AP terms efficiently

Derive and apply the sum formula by pairing.

Name or initials: ____________   Date: ____________

Reordering a finite addition does not change its sum.

1. Write a short AP both ways. For 4 + 7 + 10 + 13 + 16, reverse the order underneath.
2. Add vertically. Each vertical pair totals 20: 4 + 16, 7 + 13 and 10 + 10. There are five pairs across the two rows.
3. Remove the double count. The two rows contain twice the original sum, so 2S = 5 × 20 and S = 50.
4. Generalise. For first term a and last term l, Sₙ = n(a + l)/2 = n[2a + (n − 1)d]/2. This also works for odd n.

Worked example: Save 5, 8, 11, … units for 10 days. Last term = 5 + 9 × 3 = 32. Total = 10(5 + 32)/2 = 185 units.

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

   1. 42 · 2. 48 · 3. 36

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

   1. 116 · 2. 100 · 3. 108

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

   1. 55 · 2. 60 · 3. 65

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

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, …?

   -2 / -4 / -3

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

   5 / 6 / 7

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

   25 / 19 / 22

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

   11 / 12 / 13

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

   98 / 84 / 91

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

   36 / 45 / 54

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 / 3 / 1

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

   -3 / -5 / -4

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

   19 / 16 / 22

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

   67 / 55 / 61

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

   99 / 105 / 93

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

   164 / 148 / 156