A tile pattern grows: 1, 3, 6, … We will draw it, spot the rule, and discover the equation — then find Design 10 without drawing. Click Next.
Blue = old tiles. Orange = NEW tiles just added.
Jumps +2, +3 change, but the second jump is 1 — same. The equation will have multiplying in it, not just adding.
Take paper. Draw the next staircase. Count the tiles. Then click Next.
Drawing Design 10 would take forever. We need an equation. Watch.
tiles = n × (n + 1) ÷ 2
n wide times (n+1) tall, then cut the rectangle in half: 4 × 5 ÷ 2 = 10 ✓
First test it on a design we can count. Then jump to Design 10.
Test on Design 3 (we counted 6 tiles):
3 × 4 ÷ 2 = 12 ÷ 2 = 6 ✓ It works!
Design 10 — no drawing needed:
10 × 11 ÷ 2 = 110 ÷ 2 = 55 tiles
Design 10 is NOT Design 5 doubled. Design 5 has 15 tiles. 15 × 2 = 30. The real answer is 55! Growing patterns do not scale like that.
Second trap: guessing n². 10² = 100 — wrong. The staircase equation is n × (n+1) ÷ 2. Always test the equation on a small design you can count.
Staircase + its twin = a rectangle. n × (n+1), then cut in half.
A tile pattern grows. Design 1 has 2 tiles, Design 2 has 6, Design 3 has 12, Design 4 has 20. How many tiles are in Design 8?
Answer: (D) 72.
1. Jumps: +4, +6, +8 → not same. Subtract again: 2, 2 → same → quadratic.
2. Hunt the equation: try n × (n+1): 1×2=2 ✓, 2×3=6 ✓, 3×4=12 ✓, 4×5=20 ✓ (a full rectangle this time — no halving).
3. Design 8: 8 × 9 = 72. (Trap: 4×20 = 80 is wrong — patterns do not scale!)