Square Number Sequence

A square number is defined as a product of an integer multiplied by itself. It can also be defined as any number raised to the power 2.

For example: 169 (13 × 13), 169 is a square number. The square number sequence starts from 0 to infinity.

Mathematically, the n^th square number can be expressed as:

S_n = n²

where n is a non-negative integer (0, 1, 2, 3, 4,...).

Square Numbers can be arranged in a square grid for each square number as follows:

First few Square Numbers

There are a total of 45 square numbers between 0 to 2000. These numbers are:

0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1024, 1089, 1156, 1225, 1296, 1369, 1444, 1521, 1600, 1681, 1764, 1849, 1936

Note: Square numbers will always be positive: as ( - ) × ( - ) = ( + ) and ( + ) × ( + ) = ( + ).

Differences Between Consecutive Square Numbers

To find the difference between two consecutive square numbers you can use the formula:

(n + 1)² - n² = 2n + 1

Here’s how the differences between the first few square numbers look:

A Few things we can determine from the above table:

The difference of square of the two consecutive numbers will always be equal to the sum of numbers.

For example:

Let's take 9 and 8

The difference of Square: (9² - 8²) = 17

Sum of these two numbers: (9 + 8) = 17

The difference of the two consecutive square numbers will always be odd.

For example:

1, 3, 5, 7, 9, 11, ...

The square of an odd positive integer will always yield an odd number, while the square of an even positive integer will yield an even number.

For example:

11² = 121 (odd)

22² = 484 (even)

Related Reads,

• Perfect Squares
• Perfect Cube