Some More Interesting Patterns of Square Number

Some More Interesting Patterns of Square Number:

1. Adding triangular numbers:

If we combine two consecutive triangular numbers, we get a square number, like

2. Numbers between square numbers:

Between 1²(=1) and 2²(= 4), there are two (i.e., 2 × 1) non-square numbers 2, 3.
Between 2²(= 4) and 3²(= 9) there are four (i.e., 2 × 2) non-square numbers 5, 6, 7, 8.
Now, 3² = 9, 4² = 16.
Therefore, 4² – 3² = 16 – 9 = 7.
Between 9(= 3²) and 16(= 4²), the numbers are 10, 11, 12, 13, 14, 15 that is, six non-square numbers which is 1 less than the difference of two squares.
We have 4² = 16 and 5² = 25.
Therefore, 5² – 4² = 9.
Between 16 (= 4²) and 25(= 5²), the numbers are 17, 18,..., 24 that is, eight non-square numbers which is 1 less than the difference of two squares.

If we think of any natural number n and (n + 1), then,
(n + 1)² – n² = (n² + 2n + 1) – n² = 2n + 1.
We find that between n² and (n + 1)²  there are 2n numbers which is 1 less than the difference of two squares.
Thus, in general, we can say that there are 2n non-perfect square numbers between the squares of the numbers n and (n + 1), where n is any natural number.

3. Adding odd numbers:

The sum of first n odd natural numbers is n².

1 [one odd number] = 1 = 1²

1 + 3 [sum of first two odd numbers] = 4 = 2²

1 + 3 + 5 [sum of first three odd numbers] = 9 = 3²

If a natural number cannot be expressed as a sum of successive odd natural numbers starting with 1, then it is not a perfect square.

25 = 1 + 3 + 5 + 7 + 9. Also, 25 is a perfect square.

4. A sum of consecutive natural numbers:

We can express the square of any odd number as the sum of two consecutive positive integers ${\displaystyle \frac{n^2–1}{2}\text{and}\frac{n^2+1}{2}}$.

For Example,

11² = 121 = 60 + 61 = ${\displaystyle \frac{{11}^2–1}{2}+\frac{{11}^2+1}{2}}$.

15² = 225 = 112 + 113 = ${\displaystyle \frac{{15}^2–1}{2}+\frac{{15}^2+1}{2}}$.

5. Product of two consecutive even or odd natural numbers:

Product of two consecutive even or odd natural numbers are calculated as (a + 1) × (a – 1) = a² – 1 where a is a natural number and (a + 1), (a – 1) are two consecutive even or odd natural numbers.

For example,

29 × 31 = (30 – 1) × (30 + 1) = 30² – 1.

44 × 46 = (45 – 1) × (45 + 1) = 45² – 1.

6. Some more patterns in square numbers:

Observe the squares of numbers; 1, 11, 111,... etc. They give a beautiful pattern:

Another interesting pattern.