Chapter 1 Solutions — Patterns in Mathematics

Adding up the All 1's sequence:

1 = 1

1 + 1 = 2

1 + 1 + 1 = 3

1 + 1 + 1 + 1 = 4

This gives: 1, 2, 3, 4, 5, … — the Counting numbers sequence.

Adding the All 1's sequence up and then down:

1 = 1

1 + (1+1) + 1 = 4

1 + (1+1) + (1+1+1) + (1+1) + 1 = 9

1 + (1+1) + (1+1+1) + (1+1+1+1) + (1+1+1) + (1+1) + 1 = 16

This gives: 1, 4, 9, 16, 25, … — the Square numbers sequence.

Counting little squares in each Stacked Square shape:

Shape 1 (1×1 grid): 1 little square

Shape 2 (2×2 grid): 4 little squares

Shape 3 (3×3 grid): 9 little squares

Shape 4 (4×4 grid): 16 little squares

Shape 5 (5×5 grid): 25 little squares

Sequence: 1, 4, 9, 16, 25, … — the Square numbers sequence.

Why? The n-th shape is an n×n grid of little squares, which contains n × n = n² squares. This is exactly what square numbers represent.

Adding up the counting numbers:

1 = 1

1 + 2 = 3

1 + 2 + 3 = 6

1 + 2 + 3 + 4 = 10

1 + 2 + 3 + 4 + 5 = 15

This gives: 1, 3, 6, 10, 15, … — the Triangular numbers sequence.

Pictorial explanation: Each triangular number counts the dots in a triangular arrangement. When we add 1+2+3+...+n, we are counting dots in a triangle with n rows, which is exactly the n-th triangular number.