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Sum of first three odd numbers = 1 + 3 + 5 = 9 = 3 2

Sum of first four odd numbers = 1 + 3 + 5 + 7 = 16 = 4 2
Sum of first five odd numbers = 1 + 3 + 5 + 7 + 9 = 25 = 5 2

Sum of first n odd numbers = 1 + 3 + 5 + 7 + 9 + ... (up to n) = n 2

2
Thus, we can say that the sum of the first n odd natural numbers is n .
Pattern 3. Product of two consecutive odd natural numbers

Observe the following:
2
1 × 3 + 1 = 2 (2 is only even number between 1 and 3)
2
3 × 5 + 1 = 4 (4 is only even number between 3 and 5)
5 × 7 + 1 = 6 (6 is only even number between 5 and 7)
2
2
7 × 9 + 1 = 8 (8 is only even number between 7 and 9)
9 × 11 + 1 = 10 , etc. (10 is only even number between 9 and 11)
2
Thus, we can say that, if 1 is added to the product of two consecutive odd natural numbers, it is
equal to the square of the only even natural number between them.

Pattern 4. Product of two consecutive even natural numbers

Further, we can observe that,
2 × 4 + 1 = 9 = 3 (3 is only odd number between 2 and 4)
2
2
4 × 6 + 1 = 25 = 5 (5 is only odd number between 4 and 6)
2
6 × 8 + 1 = 49 = 7 (7 is only odd number between 6 and 8)
8 × 10 + 1 = 81 = 9 (9 is only odd number between 8 and 10)
2
10 × 12 + 1 = 121 = 11 , etc. (11 is only odd number between 10 and 12)
2
Thus, we can say that, if 1 is added to the product of two consecutive even natural numbers, it
is equal to the square of the only odd natural number between them.

Pattern 5. Squares of odd natural numbers:

Observe the following:
3 = 9 = 4 + 5 3 − 1 3 + 1
2
2
2
2
2
5 = 25 = 12 + 13 Note: Since, 3 = 9 = 4 + 5 = 2 + 2
2
7 = 49 = 24 + 25 M M
2
15 − 1 15 + 1
2
2
2
9 = 81 = 40 + 41 15 = 225 = 112 + 113 = 2 + 2
2
11 = 121 = 60 + 61 Thus, in general, 2 1 n + 1
n −
2
2
13 = 169 = 84 + 85 n = 2 + 2
2
15 = 225 = 112 + 113, etc.
2
Thus, we can say that, the square of any odd natural number, other than 1, can be expressed as
the sum of two consecutive natural numbers.
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