Lesson 12 of 14
The big picture
Show on a big screenThese lessons ask one thing. What can we be sure of?
| Idea | Remember | Example |
|---|---|---|
| A rule, not a guess | A rule must fit every join, not only the first. | 5, 8, 11 — gap 3 twice |
| Looking back | A recursive rule needs a first term to begin. | t₁ = 2, add 3 |
| Jumping there | An explicit rule gives any term from its position. | t₁₀ = 3 × 10 + 2 = 32 |
| The same step | An AP has tₙ = a + (n − 1)d. | 5, 8, 11, 14 → 32 at n = 10 |
| Adding the first n | Two copies fill a rectangle n × (n + 1). | 1 to 100 = 5050 |
| The same multiplier | A GP has tₙ = a × rⁿ⁻¹. | 5, 10, 20, 40 → 160 at n = 6 |
| A shape that multiplies | Stage n of the Sierpiński triangle has 3ⁿ triangles. | stage 4 = 81 |
| A tower and a doubling | Each size takes twice the last, plus one. | 1, 3, 7, 15, 31 |
| Does it really fit | Two rules can share three terms and part at the fourth. | 1, 2, 4 → 8 or 7 |
Open Chapter 8 now. It is called Predicting What Comes Next. Many of its ideas will feel familiar.