Patterns in Square Numbers

1. Adding Odd Numbers

There is a beautiful relationship between squares and odd numbers. Every square number is the sum of consecutive odd numbers starting from 1.

General Rule: The sum of the first $n$ odd natural numbers is $n^2$.

Visual Proof

We can visualize this by adding “layers” (L-shapes) to a square.

2. Triangular Numbers

Triangular numbers are numbers that can form a triangle: $1, 3, 6, 10, 15 \dots$. If you add two consecutive triangular numbers, you get a square number!

• $1 + 3 = 4 = 2^2$
• $3 + 6 = 9 = 3^2$
• $6 + 10 = 16 = 4^2$

3. Numbers Between Squares

How many non-square numbers lie between two consecutive squares $n^2$ and $(n+1)^2$?

Let’s check:

• Between $1^2 (1)$ and $2^2 (4)$: 2, 3 (2 numbers)
• Between $2^2 (4)$ and $3^2 (9)$: 5, 6, 7, 8 (4 numbers)
• Between $3^2 (9)$ and $4^2 (16)$: 10, 11, 12, 13, 14, 15 (6 numbers)

Pattern: There are $2n$ non-square numbers between $n^2$ and $(n+1)^2$.