Consecutive Integers Sum of Squares Calculator With Steps

Calculate the sum of squares of consecutive integers, consecutive odd integers, or consecutive even integers between two specified numbers or starting from a given integer with this versatile calculator.

Calculating the sum of squares of consecutive integers requires at least three steps:

1. List the integers that meet the conditions.
2. Calculate the square value of all integers.
3. Add up all the square values calculated in the second step, and the result is the sum of squares of consecutive integers.

Obviously, we need formulas to simplify these steps. Fortunately, we get the formula we want through the cuemath and themathpage website.

Sum of consecutive squares formula

N * (N + 1) * (2 * N + 1) / 6

Sum of squares of first N even numbers formula

2 * N * (N + 1) * (2 * N + 1) / 3

Sum of squares of first N odd numbers formula

N * (2 * N + 1) * (2 * N – 1) / 3

Obviously, these formulas are calculating the sum of squares of first N terms, that is, the starting number is 1. What if the starting number is not 1? Don’t worry, make a small change. First, calculate the sum of squares from 1 to N, then calculate the sum of squares from 1 to the starting number M, and finally subtract the sum of squares from 1 to the starting number M from the sum of squares from 1 to N. So that we can get the sum of consecutive squares from M to N.

N * (N + 1) * (2 * N + 1) / 6 – M * (M + 1) * (2 * M + 1) / 6

The same is true for even and odd numbers.

Next, let’s give some examples.

Example 1: find the sum of squares of first 20 natural numbers.

Common calculation

First 20 natural numbers are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19 and 20. Their sum of squares is

 

1² + 2² + 3² + 4² + 5² + 6² + 7² + 8² + 9² + 10² + 11² + 12² + 13² + 14² + 15² + 16² + 17² + 18² + 19² + 20²

= 1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 + 81 + 100 + 121 + 144 + 169 + 196 + 225 + 256 + 289 + 324 + 361 + 400

= 2870

Formula calculation

There are 20 integers in first 20 natural numbers. So, their sum of squares is

 

20 * (20 + 1) * (2 * 20 + 1) / 6

= 20 * 21 * 41 / 6

= 17220 / 6

= 2870

Example 2: find the sum of squares of natural numbers from 10 to 25.

Common calculation

The integers between 10 and 25 are 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25. So, Their sum of squares is

 

10² + 11² + 12² + 13² + 14² + 15² + 16² + 17² + 18² + 19² + 20² + 21² + 22² + 23² + 24² + 25²
= 100 + 121 + 144 + 169 + 196 + 225 + 256 + 289 + 324 + 361 + 400 + 441 + 484 + 529 + 576 + 625
= 5240

Formula calculation

There are 25 integers between 1 and 25, and 9 integers less than 10. So, the sum of squares of natural numbers from 10 to 25 is

 

25 * (25 + 1) * (2 * 25 + 1) / 6 – 9 * (9 + 1) * (2 * 9 + 1) / 6
= 25 * 26 * 51 / 6 – 9 * 10 * 19 / 6
= 33150 / 6 – 1710 / 6
= 5525 – 285
= 5240

Example 3: sum of squares of first 20 even numbers.

Common calculation

The first 20 even numbers are 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38 and 40. Their sum of squares is

 

2² + 4² + 6² + 8² + 10² + 12² + 14² + 16² + 18² + 20² + 22² + 24² + 26² + 28² + 30² + 32² + 34² + 36² + 38² + 40²
= 4 + 16 + 36 + 64 + 100 + 144 + 196 + 256 + 324 + 400 + 484 + 576 + 676 + 784 + 900 + 1024 + 1156 + 1296 + 1444 + 1600
= 11480

Formula calculation

There are 20 even integers between 1 and 40. So, the sum of squares of first 20 even numbers is

 

2 * 20 * (20 + 1) * (2 * 20 + 1) / 3
= 40 * 21 * 41 / 3
= 34440 / 3
= 11480

Example 4: sum of squares of even numbers from 10 to 30.

Common calculation

The even integers between 10 and 30 are 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 and 30. Their sum of squares is

 

10² + 12² + 14² + 16² + 18² + 20² + 22² + 24² + 26² + 28² + 30²
= 100 + 144 + 196 + 256 + 324 + 400 + 484 + 576 + 676 + 784 + 900
= 4840

Formula calculation

There are 15 even integers between 1 and 30, and 4 even integers less than 10. So, the sum of squares of even numbers from 10 to 30 is

 

2 * 15 * (15 + 1) * (2 * 15 + 1) / 3- 2 * 4 * (4 + 1) * (2 * 4 + 1) / 3
= 30 * 16 * 31 / 3 – 8 * 5 * 9 / 3
= 14880 / 3 – 360 / 3
= 4960 – 120
= 4840

Example 5: sum of squares of first 10 odd numbers.

Common calculation

The first 10 odd numbers are 1, 3, 5, 7, 9, 11, 13, 15, 17 and 19. Their sum of squares is

 

1² + 3² + 5² + 7² + 9² + 11² + 13² + 15² + 17² + 19²
= 1 + 9 + 25 + 49 + 81 + 121 + 169 + 225 + 289 + 361
= 1330

Formula calculation

There are 10 odd integers between 1 and 19. So, the sum of squares of first 10 odd numbers is

 

10 * (2 * 10 + 1) * (2 * 10 – 1) / 3
= 10 * 21 * 19 / 3
= 3990 / 3
= 1330

Example 6: sum of squares of odd numbers between 1 and 30.

Common calculation

The odd integers between 1 and 30 are 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27 and 29. Their sum of squares is

 

1² + 3² + 5² + 7² + 9² + 11² + 13² + 15² + 17² + 19² + 21² + 23² + 25² + 27² + 29²
= 1 + 9 + 25 + 49 + 81 + 121 + 169 + 225 + 289 + 361 + 441 + 529 + 625 + 729 + 841
= 4495

Formula calculation

There are 15 odd integers between 1 and 30. So, the sum of squares of odd numbers between 1 and 30 is

 

15 * (2 * 15 + 1) * (2 * 15 – 1) / 3
= 15 * 31 * 29 / 3
= 13485 / 3
= 4495

Whether it is a common method or a formula method, it needs to be calculated manually. Obviously, this is not suitable for everyone, especially laymen. Therefore, we have developed this online free sum of consecutive squares calculator with steps. You can easily get the answer by simply entering a few numbers. The use steps are as follows:

1. Enter the starting integer, if you don’t enter it, it starts from 1 by default.
2. Enter the ending integer or fill in the number of consecutive integers, which is required.
3. Select the type: consecutive integers, consecutive even numbers or consecutive odd numbers. The default is consecutive integers.
4. Click the calculation button.

In a blink of an eye, the answer will appear. What is even more surprising is that in addition to the answer, there will show the common calculation steps and formula calculation steps, which should be what you need urgently.