# Some more Interesting Patterns

**1) Triangular Numbers :-**

Â

Â

Triangular numbers are those numbers whose dot patterns can be arranged as triangles. Let us see what happens if we add two consecutive triangular numbers.

1 + 3 = 4 = 2^{2 }

3 + 6 = 9 = 3^{2}

6 + 10 = 16 = 4^{2}

We observe that sum of two consecutive triangular numbers is a square number.

**2. Adding Odd Numbers :-**

The squares of a natural number n are equal to the sum of the first n odd numbers.** **Thus

1^{2} = 1

2^{2} = 1 + 3 : Sum of first 2 odd numbers

3^{2} = 1 + 3 + 5 : Sum of first 3 odd numbers

8^{2} = 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 : Sum of first 8 odd numbers.

In general we can write,

n^{2} = Sum of first n odd numbers

**3.Â Difference of the squares of two consecutive numbers :-**

The difference of the squares of two consecutive numbers is equal to the sum of the numbers.

(n + 1)^{2} - n^{2} = n^{2} + 2n + 1 = (n + 1) + n

Thus 11^{2} - 10^{2 =} 121 - 100 = 21 = 11 + 10

Â 16^{2} - 15^{2 =} 256 - 225 = 31 = 16 + 15

**4. Easy method to find the square :-**

(i) 67^{2 } = 4489

(ii) 667^{2} = 444889

(iii) 6667^{2} = 44448889

Observe the following pattern, find the squares of 66667 and 666667.

Â

Â

**5. Patterns for squares of 11, 111, 1111,:-**

Â

Proceeding like this we can get the squares of 1111111^{2}, 11111111^{2} , â€¦.