# Sum of n Natural Numbers and Their Cubes

# Know About Sum of n Natural Numbers and Their Cubes

Mathematics is full of patterns. An example of such a pattern is the sum of cubes of n natural numbers. There is also different patterns involved in the **Basic Set Theory**. Numbers are exponentiated by their cubes. Three cubes are 3^{3} = 27, for example. The cubes of greater natural numbers will also be very large if we keep going.

Using less time and energy, how can we find the sum of cubes of n natural numbers? This much confusion is there to find out the **Complement of a Set**. In this guide, you will understand about simple formulas for carrying out this in this article that will ease your calculations.

## Sum of Cubes of First n Natural Numbers

A natural number is a number that starts at 1 and continues indefinitely. Add the cubes of a specific number of natural numbers starting from 1 to find the sum of cubes of first n natural numbers. To illustrate, the sum of cubes of the first 5 natural numbers can be expressed as 13 + 23 + 33 + 43 + 53, and the sum of cubes of the first 10 natural numbers as 1^{3} + 2^{3} + 3^{3} + 4^{3} + 5^{3} + 6^{3} + 7^{3} + 8^{3} + 9^{3} + 10^{3}.

Here are some examples of the sum of cubes of n natural numbers.

The sum of the cubes of the first two natural numbers is 1^{3} + 2^{3} = 1 + 8 = 9.

1 + 8 + 27 = 36 when the cubes of the first three (1^{3} + 2^{3}+3^{3}) natural numbers are added together.

The sum of the cubes of the first four natural numbers is 1^{3} + 2^{3} + 3^{3} + 4^{3} = 1 + 8 + 27 + 64 = 100.

The sum of the cubes of more natural numbers is becoming increasingly difficult to calculate as we go along. This is where a formula for the sum of cubes of n natural numbers comes into play.

What is formula?

Here is the formula for sum of cubes of n natural numbers:

sum of cubes of n natural numbers

If we have n cubes, 13 + 23 + 33 + 43 +... + n3, the formula is,

Sum (S) =

{n(n+1)2}/2

or

{n2(n+1)2}/4

Starting from 1, n represents the total number of natural numbers.

The formula for finding the sum of cubes of n natural numbers is now clear to you.

In order to memorize the formula without understanding the logic and reasoning behind it, you must understand the proof of the sum of cubes of n natural numbers. For more details on it, you must follow your regular curriculum and textbook chapters to know how this formula is derived.

Sum of cubes formula is used to find the addition of two polynomials, a^{3} + b^{3}. Here are a few examples that illustrate the sum of cubes formula. In solving algebraic expressions of various types, this factoring formula comes in handy. This formula can also be memorized within minutes. As well, it is very similar to the difference between cubes formula.

Among the most important algebraic identities is the sum of cubes formula. As a cube plus a cube, it is represented by a^{3} + b^{3}. A^{3} + b^{3} = (a + b) (a^{2} - ab + b^{2}) is the sum of cubes (a^{3} + b^{3}) formula.

So, this was the post about cubes and their sum. Hope you have found it informative.