Sum of Consecutive Cubes Calculator - Natural, Odd or Even Numbers

Welcome to the sum of consecutive cubes calculator, this calculator is used to calculate the sum of cubes of consecutive numbers within a specified range. Specify the range, which requires you to enter the starting number and ending number. A consecutive number can be a natural number, consecutive even number or consecutive odd number. For example, the sum of cubes of consecutive odd numbers between 5 and 100.

Maybe you are curious, the sum of consecutive cubes calculator is very similar to the sum of cubes of first N numbers calculator introduced earlier, how to use them? It’s very simple. If you know the number of consecutive numbers, use the sum of cubes of first N numbers calculator. If you don’t know the specific number, but only understand the range of consecutive numbers, then use the sum of consecutive cubes calculator. Of course, if you know the number of consecutive numbers and the range of consecutive numbers, it doesn’t matter, you can choose any one. You can verify whether the answers calculated by the two calculators are consistent.

To calculate the sum of cubes of continuous numbers, it is inseparable from a basic formula, that is, the formula of the sum of cubes of first N natural numbers

(n * (n + 1) / 2)² = n² * (n + 1)² / 4

According to this basic formula, formulas of the sum of cubes of first N even numbers and first N odd numbers can be derived

The sum of cubes of first n even numbers numbers

2 * (n * (n + 1))² = 2 * (n² * (n + 1)²)

The sum of cube of first n odd numbers numbers

n² * (2 * n² – 1)

Please refer to the following URL for the derivation process

https://www.geeksforgeeks.org/sum-cubes-even-odd-natural-numbers/

Next, to calculate the sum of cubes of consecutive numbers within the specified range, we can use the sum of cubes of natural numbers between 1 and the ending number, subtract the sum of cubes of natural numbers from 1 to starting number before. For example, the formula for the sum of cubes of consecutive nature number between M and N is as follows:

N² * (N + 1)² / 4 – M² * (M – 1)² / 4

Similarly, we can get the formula for the sum of cubes of consecutive even numbers and consecutive odd numbers.

The sum of cubes of consecutive even numbers between M and N formula

2 * b² * (b + 1)² – 2 * a² * (a + 1)²

Please note that b in the formula represents the number of even integers from 1 to M, and a represents the number of even integers from 1 to N – 1.

The sum of cubes of consecutive odd numbers between M and N formula

b² * (2 * b² – 1) – a² * (2 * a² – 1)

Please note that b in the formula represents the number of odd integers from 1 to M, and a represents the number of odd integers from 1 to N – 1.

M represents the starting number to be input by the sum of consecutive cubes calculator, and N is the ending number.

The starting number is 1, so we can use the sum of cubes of first N natural numbers to calculate it. In here, N is equal to 20.

n² * (n + 1)² / 4

= 20² * (20 + 1)² / 4

= 20² * 21² / 4

= 400 * 441 / 4

= 44100

Use the sum of consecutive cubes calculator to find the sum of cubes of consecutive numbers from 1 to 20.  Select nature numbers, and input 1 in the start number box, input 20 in the end number box, then click calculate button.

The result is 44100, which is consistent with the calculation by the above formula.

The even numbers between 1 and 20 are 2, 4, 6, 8, 10, 12, 14, 16, 18 and 20. There are 10 numbers in total. So we can use the sum of cubes of first N even numbers to calculate it. In here, N is equal to 10.

2 * (n² * (n + 1)²)

= 2 * (10² * (10 + 1)²)

= 2 * (10² * 11²)

= 2 * 100 * 121

= 24200

Use the sum of consecutive cubes calculator to find the sum of cubes of consecutive even numbers from 1 to 20.  Select even integers, and input 1 in the start number box, input 20 in the end number box, then click calculate button.

If shown, the result is also 24200.

The odd numbers between 10 and 50 are 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49. There are 20 numbers in total. The odd numbers before 10 are 1, 3, 5, 7, 9, and 5 in total. So, there are 25 odd numbers from 1 to 50. Next, use the sum of cubes of consecutive odd numbers between M and N formula to calculate the answer

b² * (2 * b² – 1) – a² * (2 * a² – 1)

= 25² * (2 * 25² – 1) – 5² * (2 * 5² – 1)

= 25² * (2 * 625 – 1) – 5² * (2 * 25 – 1)

= 625 * (1250 – 1) – 25 * (50 – 1)

= 625 * 1249 – 25 * 49

= 780625 – 1225

= 779400

Use the sum of consecutive cubes calculator to find the sum of cubes of consecutive odd numbers from 10 to 50.  Select odd integers, and input 10 in the start number box, input 50 in the end number box, then click calculate button.

If shown, the result is also 779400.