#### description

- Adding triangular numbers
- Numbers between square numbers
- Adding odd numbers
- Sum of consecutive natural numbers
- Product of two consecutive even or odd natural numbers
- Some more patterns in square numbers

#### notes

**1. Adding triangular numbers : **

The triangular number means numbers whose dot patterns can be arranged as triangles.

If we combine two consecutive triangular numbers , we get a square numbers.

**2. Numbers between square numbers :**

Let us now see if we can find some interesting pattern between two consecutive square numbers.

Between 12(=1) and 22(= 4) there are two (i.e., 2 × 1) non square numbers 2, 3.

We have `4^2 = 16` and ` 5^2 = 25`

Therefore, `5^2 – 4^2 = 9`

Between `16(= 4^2)` and `25(= 5^2)` the numbers are 17, 18, ... , 24 that is, eight non square numbers which is 1 less than the difference of two squares.

Consider `7^2` and `6^2`. Can you say how many numbers are there between `6^2` and `7^2`?

If we think of any natural number n and (n + 1), then,

`(n + 1)^2 – n^2 = (n^2 + 2n + 1) – n^2 = 2n + 1`.

We find that between `n^2` 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).

**3. Adding odd numbers **

Consider the following

1 [one odd number] = 1 = `1^2`

1 + 3 [sum of first two odd numbers] = 4 = `2^2`

1 + 3 + 5 [sum of first three odd numbers] = 9 = `3^2`

1 + 3 + 5 + 7 [... ] = 16 = `4^2`

1 + 3 + 5 + 7 + 9 [... ] = 25 = `5^2`

1 + 3 + 5 + 7 + 9 + 11 [... ] = 36 =`6^2`

So we can say that the sum of first n odd natural numbers is `n^2`. Looking at it in a different way, we can say: ‘If the number is a square number, it has to be the sum of successive odd numbers starting from 1.

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.

**4. A sum of consecutive natural numbers **

Consider the following

`9^2 = 81 = 40 + 41`

`11^2 = 121 = 60 + 61`

`15^2 = 225 = 112 + 113`

We can express the square of any odd number as the sum of two consecutive positive integers.

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

11 × 13 = 143 = `12^2` – 1

Also 11 × 13 = (12 – 1) × (12 + 1)

Therefore, 11 × 13 = (12 – 1) × (12 + 1) = `12^2` – 1

Similarly, 13 × 15 = (14 – 1) × (14 + 1) = `14^2` – 1

29 × 31 = (30 – 1) × (30 + 1) = `30^2` – 1

44 × 46 = (45 – 1) × (45 + 1) = `45^2` – 1

So in general we can say that (a + 1) × (a – 1) = `a^2` – 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 .