You can add a series of consecutive odd numbers manually, but there's an easier way, especially if you're working with a lot of numbers. Once you've mastered this simple formula, you can perform these calculations without the help of a calculator. There is also a simple way to find a series of consecutive odd numbers from their sum.
Step
Part 1 of 3: Applying the Formula to Add Up a Sequential Series of Odd Numbers
Step 1. Select an endpoint
Before you begin, you need to determine the last number of the series that you want to calculate. This formula helps you add up any sequence of odd numbers, starting with 1.
If you do the problem, this number will be given. For example, if the question asks you to find the sum of all consecutive odd numbers between 1 and 81, your endpoint is 81
Step 2. Add up by 1
The next step is to add the endpoint number by 1. Now, you get the even number needed for the next step.
For example, if your endpoint is 81, it means 81 + 1 = 82
Step 3. Divide by 2
Once you get an even number, divide by 2. This way you get an odd number equal to the number of digits added together.
For example, 82/2 = 41
Step 4. Square the result
Finally, you need to square the result of the previous division, by multiplying the number by itself. If so, you've got the answer.
For example, 41 x 41 = 1681. That is, the sum of all consecutive odd numbers between 1 and 81 is 1681
Part 2 of 3: Understanding How Formulas Work
Step 1. Notice the pattern
The key to understanding this formula lies in the underlying pattern. The sum of all consecutive odd-number sets starting with 1 is always equal to the square of the number of digits of the numbers added together.
- Sum of the first odd numbers = 1
- The sum of the first two odd numbers = 1 + 3 = 4 (= 2 x 2).
- The sum of the first three odd numbers = 1 + 3 + 5 = 9 (= 3 x 3).
- The sum of the first four odd numbers = 1 + 3 + 5 + 7 = 16 (= 4 x 4).
Step 2. Understand the interim data
By solving this problem, you learn more than adding up numbers. You also learn how many consecutive digits are added together, which is 41! This is because the number of digits added is always equal to the square root of the sum.
- The sum of the first odd numbers = 1. The square root of 1 is 1, and only one digit is added.
- The sum of the first two odd numbers = 1 + 3 = 4. The square root of 4 is 2, and the two digits add up.
- The sum of the first three odd numbers = 1 + 3 + 5 = 9. The square root of 9 is 3, and the three digits add up.
- The sum of the first two odd numbers = 1 + 3 + 5 + 7 = 16. The square root of 16 is 4, and there are four digits added together.
Step 3. Simplify the formula
Once you understand the formula and how it works, write it down in a format that can be used with any number. The formula for finding the sum of the first odd numbers is n x n or n squared.
- For example, if you plug 41 into, you get 41 x 41, or 1681, which is the sum of the first 41 odd numbers.
- If you don't know how many numbers to work with, the formula to find the sum between 1 and is (1/2(+ 1))2
Part 3 of 3: Determining Sequential Odd Number Series from Summing Results
Step 1. Understand the difference between the two types of questions
If you are given a series of consecutive odd numbers and asked to find their sum, we recommend using the formula (1/2(+ 1))2. On the other hand, if the question gives you a summed number, and asks you to find a sequence of consecutive odd numbers that produces that number, the formula that needs to be used is different.
Step 2. Make n the first number
To find a series of consecutive odd numbers whose sum matches the number given the problem, you need to create an algebraic formula. Start by using as a variable the first number in the series.
Step 3. Write down the other numbers in the series using the variable n
You need to determine how to write the other numbers in the series with the variable. Since they are all odd numbers, the difference between the numbers is 2.
That is, the second number in the series is + 2, and the third is + 4, and so on
Step 4. Complete the formula
Now that you know the variable that represents each number in the series, it's time to write down the formula. The left side of the formula must represent the numbers in the series, and the right side of the formula represents the sum.
For example, if you were asked to find a series of two consecutive odd numbers that add up to 128, the formula would be + + 2 = 128
Step 5. Simplify the equation
If there is more than one on the left side of the equation, add them all together. Thus, the equation is easier to solve.
For example, + + 2 = 128 simplifies to 2n + 2 = 128.
Step 6. Isolate n
The final step to solving the equation is to make it a single variable on one side of the equation. Remember, all changes made on one side of the equation must also occur on the other side.
- Calculate addition and subtraction first. In this case, you need to subtract 2 from both sides of the equation to get as a single variable on one side. Therefore, 2n = 126.
- Then, do multiplication and division. In this case, you need to divide both sides of the equation by 2 to isolate so that = 63.
Step 7. Write down your answers
At this point, you know that = 63, but the work is still not done. You still have to make sure that the questions in the questions have been answered. If the question asks for a series of consecutive odd numbers, write down all the numbers.
- The answer to this example is 63 and 65 because = 63 and + 2 = 65.
- We recommend that you check your answers by entering the calculated numbers into the questions. If the numbers don't match, try working again.
