Linear Patterns: Discovering the Constant Step
Watch a tile pattern grow, count the new tiles, and discover the formula 2n − 1 hiding in the picture
The growing L-staircase
Look at the four stages of the tile pattern:
- Stage 1: 1 square.
- Stage 2: 3 squares (an L-shape).
- Stage 3: 5 squares.
- Stage 4: 7 squares.
The sequence of counts is — the odd numbers.
How is each stage built from the previous one? Look carefully: each stage takes the previous shape and adds exactly 2 squares — one on top of the vertical column, one on the right of the horizontal row. The pattern grows in a steady L-shape rhythm, and the count goes up by 2 every stage.
From "next is +2" to "the nth term is $2n - 1$"
Knowing each stage is 2 more than the previous lets us continue the sequence: . But this is not enough by itself — to find stage 100, we'd still have to count 100 jumps of 2.
We want a formula that gives the count directly in terms of the stage number . Let's hunt for it:
| Stage | Count | Pattern |
|---|---|---|
| 1 | 1 | ✓ |
| 2 | 3 | ✓ |
| 3 | 5 | ✓ |
| 4 | 7 | ✓ |
| 5 | 9 | ✓ |
The rule is count = . This is a linear polynomial in : degree 1, leading coefficient 2, constant term .
The constant difference of 2 between consecutive counts and the leading coefficient 2 in the formula are the same number — that's no coincidence. The leading coefficient of a linear polynomial is exactly the per-step change.
The growing L-staircase
Look at the four stages of the tile pattern:
- Stage 1: 1 square.
- Stage 2: 3 squares (an L-shape).
- Stage 3: 5 squares.
- Stage 4: 7 squares.
The sequence of counts is — the odd numbers.
How is each stage built from the previous one? Look carefully: each stage takes the previous shape and adds exactly 2 squares — one on top of the vertical column, one on the right of the horizontal row. The pattern grows in a steady L-shape rhythm, and the count goes up by 2 every stage.
From "next is +2" to "the nth term is $2n - 1$"
Knowing each stage is 2 more than the previous lets us continue the sequence: . But this is not enough by itself — to find stage 100, we'd still have to count 100 jumps of 2.
We want a formula that gives the count directly in terms of the stage number . Let's hunt for it:
| Stage | Count | Pattern |
|---|---|---|
| 1 | 1 | ✓ |
| 2 | 3 | ✓ |
| 3 | 5 | ✓ |
| 4 | 7 | ✓ |
| 5 | 9 | ✓ |
The rule is count = . This is a linear polynomial in : degree 1, leading coefficient 2, constant term .
The constant difference of 2 between consecutive counts and the leading coefficient 2 in the formula are the same number — that's no coincidence. The leading coefficient of a linear polynomial is exactly the per-step change.