Welcome to the sum of cubes of first N numbers calculator. There are three cases for the first N numbers. One is the first N natural numbers, the other is the first N even numbers, and the third is the first N odd numbers. This calculator is to calculate the sum of cubes of first N numbers with steps. Since the cube value of numbers is generally larger, it is more difficult to calculate their cube sum. So, I believe this calculator can help you.

# Sum of Cubes of First N Natural, Odd or Even Numbers Calculator

## Sum of Cubes of First N Number Calculator with Steps

As you can see, the use of this calculator is very simple, just enter the value of N, then select the number type, and click the calculation button. You may notice that there is also a start field, which defaults to 1 without input, that is, the first N numbers calculated from 1. Of course, if you want to calculate from 5, then you should enter 5 here, which means that calculate the sum of cubes of first N numbers starting from 5, such as 5,6,7,8… The calculator will give the answer in a blink of an eye, which is very convenient.

## How to find the sum of cubes of first N numbers?

Maybe you are interested in the realization process of this calculator. Here are three important formulas involved in the realization of this calculator.

### The sum of cubes of first N natural numbers formula

(n * (n + 1) / 2)

^{2}= n^{2}* (n + 1)^{2}/4

### The sum of cubes of first N even numbers formula

2 * (n * (n + 1))

^{2}= 2 * (n^{2}* (n + 1)^{2})

### The sum of cube of first N odd numbers formula

n

^{2}* (2 * n^{2}– 1)

The N in the formula is the value of N that the calculator needs to enter.

If the starting number is not 1, but other numbers, such as 5, then we can modify the formula. First, calculate the sum of cubes of first N + 5 numbers, and then subtract the sum of cubes of natural numbers before 5, and finally get the result we want. Of course, if you are interested in the proof process of the formulas, you can refer to the following two pages

__https://www.cuemath.com/algebra/sum-of-cubes-of-n-natural-numbers/__

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

## Solved examples using sum of cubes of first n number calculator

### Example 1: Find the sum of cubes of first 12 natural numbers.

Use the formula to calculate the sum of cubes of first 12 natural numbers

The formula is

n^{2}* (n + 1)^{2}/ 4

12

^{2}* (12 + 1)^{2}/ 4

= 144 * 13^{2}/ 4

= 144 * 169 / 4

= 24336 / 4

= 6084

Use the sum of cubes of first N number calculator to find the sum of cubes of first 12 natural numbers.

So, the sum of cubes of first 12 natural numbers is 6084.

### Example 2: What is the sum of cubes of first 10 even numbers?

Use the formula to calculate the sum of cubes of first 10 even numbers

The formula is

2 * (n^{2}* (n + 1)^{2})

2 * (10

^{2}* (10 + 1)^{2})

= 2 * (10^{2}* 11^{2})

= 2 * 100 * 121

= 24200

Use the sum of cubes of first N number calculator to find the sum of cubes of first 10 even numbers.

So, the sum of cubes of first 10 even numbers is 24200.

### Example 3: What is the sum of cubes of first 20 odd numbers?

Use the formula to calculate the sum of cubes of first 20 odd numbers

The formula is

n^{2}* (2 * n^{2}– 1)

20

^{2}* (2 * 20^{2}– 1)

= 400 * (2 * 400 – 1)

= 400 * (800 – 1)

= 400 * 799

= 319600

Use the sum of cubes of first N number calculator to find the sum of cubes of first 20 odd numbers.

So, the sum of cubes of first 20 odd numbers is 319600.

### Example 4: Find the sum of cubes of first 8 natural numbers starting from 5.

The first 8 natural numbers starting from 5 are 5, 6, 7, 8, 9, 10, 11, 12.There are 4 nature numbers before 5. So, there are 12 natural numbers starting from 1. Use the formula to calculate the sum

12

^{2}* (12 + 1)^{2}/ 4 – 4^{2}* (4 + 1)^{2}/ 4

= 12^{2}* 13^{2}/ 4 – 4^{2}* 5^{2}/ 4

= 144 * 169 / 4 – 16 * 25 / 4

= 24336 / 4 – 400 / 4

= 6084 – 100

= 5984

Use the sum of cubes of first N number calculator to find the sum of cubes of first 8 natural numbers starting from 5.

Thus, the sum of cubes of first 8 natural numbers starting from 5 is 5984.

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