Some Interesting Patterns

1. Adding consecutive odd numbers

Observe the following pattern of sums of odd numbers.

Is it not interesting?

How many consecutive odd numbers will be needed to obtain the sum as 103?

Express the following numbers as the sum of odd numbers using the above pattern?

Instructions

(i) 63

63 = = 31 + + + + +

Starting from = 53, we take the sum of the next six odd natural numbers i.e. 31,33,35,37,39 and 41.

(ii) 83

83 = = 57 + + + + + + +

Starting from = 73, we take the sum of the next eight odd natural numbers i.e. 57,59, 61, 63, 65, 67, 69 and 71.

(iii) 73

73= = 43 + + + + + +

Starting from = 63, we take the sum of the next seven odd natural numbers i.e. 43,45, 47, 49, 51, 53 and 55.

Consider the following pattern:

23−13 = 1 + 2 × 1 × 3

33 – 23 = 1 + 3 × 2 × 3

43 – 33 = 1 + 4 × 3 × 3

Using the above pattern, find the value of the following:

Instructions

(i) 73 - 63

73 – 63 = 1 + × × 3

We find the product of the two bases (i.e. 7 and 6) and further multiply the product with 3 and add 1 to it.

(ii) 123 - 113

123 – 113 = 1 + × × 3

We find the product of the two bases (i.e. 12 and 11) and further multiply the product with 3 and add 1 to it.

(iii) 203 - 193

203 – 193 = 1 + × × 3

We find the product of the two bases (i.e. 20 and 19) and further multiply the product with 3 and add 1 to it.

(iv) 513 - 503

513 – 503 = 1 + × × 3

We find the product of the two bases (i.e. 51 and 50) and further multiply the product with 3 and add 1 to it.

Chapter 6: Square Roots and Cube Roots