TwiddleLab / Math / Counting and Patterns

Sequences: Spot the Rule

Lesson 5 of 11 · 7 min · Free, no account needed

What you will learn

• An arithmetic sequence adds the same amount each step: 4, 7, 10, 13... Term n is first + (n - 1) × jump.
• A geometric sequence multiplies by the same number each step: 1, 2, 4, 8... Term n is first × ratio to the power n - 1.
• Adding makes a straight ramp. Multiplying makes a rocket that always wins in the end.
• Some patterns are neither, like squares or Fibonacci.

Start the lesson

How the lesson goes

1. Plan A: +$10 a day. Plan B: 1 cent that doubles every day.

   Plan A has $300 on day 30. How much does Plan B have? Drag to your bet.

2. A tile pattern grows. Build stage 4.

   Your board starts as a copy of stage 3. Tap cells to add or remove tiles, then Check.

   Each stage adds the same 3 tiles, so the counts go 4, 7, 10, 13.

3. This pattern grows by copying itself.

   Drag the light copy onto the outline to make the next stage. Twice.

4. Two kinds of sequence.

   Arithmetic: add the same jump d. Term n is a₁ + (n - 1) × d: 4, 7, 10, 13 has d = 3.

   Geometric: multiply by the same ratio r. Term n is a₁ × r to the power n - 1: 1, 2, 4, 8 has r = 2.

   To tell them apart, look at the jumps, then at the ratios.

5. Sort the sequences by their rule.

   Drag each one into add the same, multiply the same, or neither.

6. Write the rule for the tile pattern.

   Tiles at stage n = [ ] × n + [ ]. Then your rule draws stage 10.

   Every stage adds 3, so the rule is 3 × n plus a fixed part. At n = 1 there are 4 tiles, so the fixed part is 1.

7. When does Plan B pass Plan A?

   Slide through the days. Find the first day Plan B has more money.

   Plan B on day n is 0.01 × 2 to the power n - 1. On day 15 that is 0.01 × 16,384 = $163.84, more than Plan A's $150.

8. A loop that doubles.

   Drag the pin to the number this program prints, then Run.

   Starting at 1 and doubling 5 times gives 2 to the power 5 = 32.

9. Game: tower guess. How tall is the stage far down the line?

   One guess per tower. Exact is 3 points, within 10% is 2.

10. Adding makes a ramp. Multiplying makes a rocket.

    Race the two plans to day 30.

Previous lesson: Pascal's Triangle · Next lesson: Adding Up a Sequence · Part of the course: Counting and Patterns

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