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Observe the pattern. Understand the interim data. Generalize the formula.

Article:
The key to understanding this formula is to recognize the underlying pattern. The sum of any set of consecutive odd numbers starting with 1 is always equal to the square of the number of digits that were added together.  Sum of first odd number = 1 Sum of first two odd numbers = 1 + 3 = 4 (= 2 x 2). Sum of first three odd numbers = 1 + 3 + 5 = 9 (= 3 x 3). Sum of first four odd numbers = 1 + 3 + 5 + 7 = 16 (= 4 x 4). By solving this problem, you learned more than the sum of the numbers. You also learned how many consecutive digits were added together: 41! This is because the number of digits added together is always equal to the square root of the sum.  Sum of first odd number = 1. The square root of 1 is 1, and only one digit was added. Sum of first two odd numbers = 1 + 3 = 4. The square root of 4 is 2, and two digits were added. Sum of first three odd numbers = 1 + 3 + 5 = 9. The square root of 9 is 3, and three digits were added. Sum of first four odd numbers = 1 + 3 + 5 + 7 = 16. The square root of 16 is 4, and four digits were added. Once you understand the formula and how it works, you can write it down in a format that will be applicable no matter what numbers you are dealing with. The formula to find the sum of the first n odd numbers is n x n or n squared.  For example, if you plugged 41 in for n, you would have 41 x 41, or 1681, which is equal to the sum of the first 41 odd numbers. If you don't know how many numbers you are dealing with, the formula to determine the sum between 1 and n is (1/2(n + 1))2