What two consecutive integers sum of squares is 613?

The sum of squares of two consecutive integers is 613. What are these two consecutive numbers? The answer is 17 and 18. You must be interested in how to find these two consecutive integers whose sum of squares is 613. Here will introduce a calculator and 2 methods to solve this problem. The calculator is suitable for everyone, including laymen. Two methods are suitable for those who have some foundation in mathematics. let’s start.

1. Sum-Squares-Based 2 Consecutive Integers Calculator

Result:
Verify:

With the help of the above calculator, we can easily find 2 consecutive integers whose sum of squares is 613. The specific steps are as follows:

  • 1Enter 613 into the input box.
  • 2Click the calculation button.

In the blink of an eye, the answer will appear. As shown in the figure, there are two sets of answers, one set is 17 and 18, and the other set is -18 and -17.

What two consecutive integers sum of squares is 613

Very simple, if you have other similar problems: find 2 consecutive integers based on the sum of squares. It can be calculated by the sum-squares-based 2 consecutive integers calculator or more advanced sum squares based consecutive integers calculator.

2. Hypothetical method

Assuming that N is used to represent the first integer, then the second integer can be represented by N + 1. Now, the sum of squares of 2 consecutive integers is 613, which can be expressed by the equation

N2 + (N + 1)2 = 613

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

N2 + (N + 1)2 = 613

N2 + N2 + 2 * N + 1 = 613

2 * N2 + 2 * N + 1 = 613

2 * N2 + 2 * N + 1 – 613 = 0

2 * N2 + 2 * N – 612 = 0

N2 + N – 306 = 0

(N – 17) * (N + 18) = 0

N = 17 or N = -18

So, the value of the first integer is 17 or -18, then the second integer is N + 1 = 18 or -17.

Obviously, there are 2 sets of answers for which the sum of the squares of two consecutive numbers is 613. The positive integers are 17 and 18, the negative integers are -18 and -17. It is consistent with the answer calculated by the calculator in the first method!

3. Average method

1
Calculate the average of the sum of squares first.

That is, divide the sum of squares by the number. Here it is 613 / 2 = 306.5.

2
Then calculate the square root of the average.

Here it is √306.5 = 17.493.

3
Finally, find consecutive integers near the square root value.

In here, find 2 consecutive integers around 17.493, and their average value is equal to 17.493.

Through the above 3 steps, you can find that these 2 consecutive integers are 17 and 18.

Let us verify that the answer is correct?

172 + 182 = 289 + 324 = 613

Obviously, this answer is correct.

Next, we consider the negative forms of these two integers -18 and -17, and we can get another answer.

Problems can be solved by two consecutive integers whose sum of squares is 613

Now, we have found 2 consecutive integers whose sum of squares is 613. At the same time, the following problems can also be solved incidentally.

  • 1The sum of squares of two consecutive integers is 613, and their sum is 17 + 18 = 35.
  • 2The sum of squares of two consecutive integers is 613, and their product is 17 * 18 = 306.
  • 3The sum of squares of two consecutive integers is 613, and the sum of their cubes is 173 + 183 = 10745.
  • 4The sum of squares of two consecutive integers is 613, and the smaller positive integer is 17.
  • 5The sum of squares of two consecutive integers is 613, and the larger positive integer is 18.
  • 6The sum of squares of two consecutive integers is 613, and their average is 17.5.

Of course, in addition to the three methods described above for finding two consecutive integers whose sum of squares is 613, there are other methods. Have you encountered them? If so, please leave a message to tell us, thank you!

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