Sum-Squares-Based Consecutive Integers Calculator

Consecutive integers calculator based on the sum of squares

Consecutive Integers Result:
Sum of Squares Verify:

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The consecutive integers calculator based on the sum of squares allows you to find a specified number of consecutive integers, even or odd numbers, and make their sum of squares equal to the given sum of squares. This is just the opposite of calculating the sum of squares of consecutive integers, but the difficulty has increased a lot. Maybe you can easily calculate the sum of squares of consecutive integers, but you may be stumped by the problem of finding consecutive integers based on the given sum of squares. Therefore, a free calculator for finding consecutive integers based on the sum of squares is necessary.

Next, let’s look at the components of this calculator. Obviously, at least two parts are needed, one is the value of the sum of squares, and the other is the number of consecutive integers. In addition, we have also added a type section, whose value contains three types: consecutive integers, consecutive even numbers, and consecutive odd numbers to meet the needs of different calculations.

How to use sum-squares-based consecutive integers calculator

The sum-squares-based consecutive integers calculator is very simple to use. It can be completed in a few steps. Whether you are a professional or non-professional, you can easily use it. The specific steps are as follows:

  1. Enter the sum of squares, which is usually a positive integer.
  2. Enter the number of consecutive integers.
  3. Select the type. The default is consecutive integers. If necessary, you can choose consecutive even integers or consecutive odd integers.
  4. Click the calculation button.

The calculation result will be presented immediately. In addition, the calculator has a verification process to ensure the results are correct.

Manual calculation

Of course, using the sum-squares-based consecutive integers calculator is very convenient, but if you want to calculate it manually, there is no problem. Here are two common methods. The two methods are explained in detail below with the same example: find two consecutive positive integers sum of whose squares is 365.

Method 1: Hypothetical Method

Assuming that N is used to represent the first integer, then the second integer can be represented by N + 1, so two consecutive positive integers sum of whose squares is 365 can be represented by an equation

N2 + (N + 1)2 = 365

This is a quadratic equation in one variable, we can solve this equation and get the value of the first integer.

N2 + (N + 1)2 = 365

N2 + N2 + 2 * N + 1 = 365

2 * N2 + 2 * N + 1 = 365

2 * N2 + 2 * N + 1 – 365 = 365 – 365

2 * N2 + 2 * N – 364 = 0

N2 + N – 182 = 0

(N – 13) * (N + 14)  = 0

N = 13 or N = -14

Therefore, the value of the first integer is 13 or -14, then the second integer is N + 1 = 14 or -13.

The answer came out, these two positive integers are 13 and 14, negative integers are -14 and -13.

This is a relatively common method. The key is to solve the quadratic equation. If the number of consecutive integers to be calculated is greater than 2, instead of 2, the calculation process will be more complicated, but the overall idea is still the same.

Method 2: Average method

Putting aside these two consecutive integers, we use their average A to replace them. Therefore, the one-dimensional quadratic equation of the first method can be expressed as

A2 + A2 = 365

2 * A2 = 365

A2 = 365 / 2

A2 = 182.5

Then, use the root calculator to easily calculate out A = 13.5.

Because A is the average of these two consecutive integers, it can be calculated that these two consecutive integers are 13 and 14.

Verification: 132 + 142 = 365. The result is correct.

Summarize the calculation process of the average method:

  1. Divide the sum of squares by the number of consecutive integers to get the square of the average.
  2. Use the root calculator to calculate the square of the average to get the average.
  3. Find consecutive integers that meet the conditions around the average.

This method is much more convenient than the first method, but requires a root calculator to assist.

Conclusion

Well, these are two more common methods for calculating consecutive integers based on the sum of squares. Of course, there are others. If you encounter them, you can leave us a message. We will add some good methods to the article.

Finally, it is recommended to use our free sum-squares-based consecutive integers calculator. Compared with manual calculation, our calculator has many advantages. The calculation time is short, the result is accurate, and it also comes with a verification function. In addition to consecutive integers, it can also calculate consecutive even numbers or consecutive odd numbers. Welcome to use it.

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