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