You can add up a series of consecutive odd numbers by hand, but there is a much easier method of doing this, especially if you have a lot of digits to add up. Once you learn a simple formula, you will be able to add up these numbers very quickly without using a calculator. Also there is a very easy way to calculate which consecutive numbers give a specific sum.
Steps
Part 1 of 3: Applying the Formula for Summing a Sequence of Consecutive Odd Numbers
Step 1. Choose an end point
Before starting, you need to decide what will be the last consecutive issue in the series. This formula can help you add any series of consecutive odd numbers, starting with 1.
If you have a task, this number will be assigned to you. For example, if a problem asks you to find the sum of all consecutive odd numbers between 1 and 81, the final number is 81
Step 2. Add 1
The next step is to simply add 1 to the final number. You should get an even number, which is crucial for the next step.
For example, if the final number is 81, 81 + 1 = 82
Step 3. Divide by 2
Once you have an even number, you should divide it by 2. You will get an odd value equal to the number of digits added together.
For example, 82/2 = 41
Step 4. Square the sum
The last step is to calculate the square of the number, or multiply it by itself. Once done, you will get the result.
For example, 41 x 41 = 1681. This means that the sum of all consecutive odd numbers between 1 and 81 is 1681
Part 2 of 3: Understanding How the Formula Works
Step 1. Observe the repeating pattern
The secret to understanding this formula is to recognize the underlying pattern. The sum of any series of consecutive odd numbers starting from 1 is always equal to the square of the number of digits added together.
- Sum of the first odd number = 1.
- Sum of the first two odd numbers = 1 + 3 = 4 (= 2 x 2).
- Sum of the first three odd numbers = 1 + 3 + 5 = 9 (= 3 x 3).
- Sum of the first four odd numbers = 1 + 3 + 5 + 7 = 16 (= 4 x 4).
Step 2. Understand the partial data
By solving this problem, you learned more than the sum of the numbers. You also figured out how many consecutive digits were added: 41! This is because the number of digits added together is always equal to the square root of the sum.
- The sum of the first odd number = 1. The square root of 1 is 1 and only one number has been added.
- The sum of the first two odd numbers = 1 + 3 = 4. The square root of 4 is 2 and two digits have been added together.
- The sum of the first three odd numbers = 1 + 3 + 5 = 9. The square root of 9 is 3 and three digits have been added together.
- The sum of the first four odd numbers = 1 + 3 + 5 + 7 = 16. The square root of 16 is 4 and four digits have been added.
Step 3. Generalize the formula
Once you understand the formula and how it works, you can write it in an applicable format regardless of the numbers you are dealing with. The formula for calculating the sum of the first odd numbers is n x n or n squared.
- For example, if you substitute 41 a, you would have 41 x 41, or 1681, which is the sum of the first 41 odd numbers.
- If you don't know how many numbers you are dealing with, the formula for determining the sum between 1 and is (1/2 (+ 1))2.
Part 3 of 3: Determine Which Consecutive Odd Numbers Give a Certain Sum
Step 1. Learn the differences between the two types of problems
If you are given a series of consecutive odd numbers and asked to calculate their sum, you should use the equation (1/2 (+ 1))2. If, on the other hand, you are assigned a sum and you are asked to find the series of consecutive odd numbers that compose it, you must use a different formula.
Step 2. Match n to the first number
To find out which consecutive odd numbers give a specific sum, you need to create an algebraic formula. Start by using to represent the first number in the sequence.
Step 3. Write the remaining numbers in relation to n
You need to determine how to write the other numbers in the sequence relative to. Since these are consecutive odd numbers, the difference between two successive numbers will always be 2.
This means that the second number in the series will be + 2, the third + 4, etc
Step 4. Complete the formula
Once you know how to represent all the numbers in the series, it's time to write the formula. The left part must represent the numbers of the series, the right part their sum.
For example, if you are asked to find a series of two consecutive odd numbers whose sum equals 128, you should write + + 2 = 128
Step 5. Simplify the equation
If there is more than one term with on the left side, add them together. This will make it much easier to fix the problem.
For example, + + 2 = 128 simplifies to 2n + 2 = 128.
Step 6. Island n
The last step in solving the equation is to isolate one side of the equation. Remember that any changes you make on one side of the equation must be repeated on the other side as well.
- Solve addition and subtraction first. In this case you have to subtract 2 from both sides of the equation to get it alone, then 2n = 126.
- Move on to multiplications and divisions. In this case you have to divide both sides of the equation by 2, if you want to isolate, then = 63.
Step 7. Write your answer
At this point you know that = 63, but you are not done yet. You need to make sure you fully answer the question that has been asked of you. If you are asked which series of consecutive odd numbers gives a certain sum, you have to write down all the numbers that make it up.
- The answer to this problem is 63 and 65, because = 63 and + 2 = 65.
- It is always a good idea to check the solution by substituting the numbers in the equation. If you don't get the desired amount as a result, try doing the math again.
