a) We have to understand the pattern of odd numbers sequence in order to prove their sum. The total numbers of any set of the ordered odd numbers that are beginning with 1 are always equal to the square of their number of digits. Those are added together. If 1, 3, 5, 7, 9, 11, …, (2n - 1) are the odd numbers, then we can sayThe sum of the first odd number is equal to 1.Sum of the first two odd numbers is equal to 1 + 3 = 4 (4 = 2 x 2).Sum of the first three odd numbers is equal to 1 + 3 + 5 = 9 (9 = 3 x 3).Sum of the first four odd numbers is equal to 1 + 3 + 5 + 7 = 16 (16 = 4 x 4).b) The number of digits that are added collectively in an ordered manner is always equal to the square root of the total number.The sum of the first odd number is equal to 1.The square root of 1, √1 = 1, so from here we understand that only one digit was added.Sum of consecutive two odd numbers is equal to 1 + 8 = 9.The square root of 9, √9 = 3, so here also we can see two digits were added.Hence, from the above-proved sums, we can prove the formula. In order to find the sum of the first n-odd numbers is n x n or n2.For example, if we put n = 12 and then we have 12 x 12 = 144. This is equal to the sum of the first 12 odd numbers.