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Sum of the Cubes of the First n Natural Numbers

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  • 0:00 Sum of the First N…
  • 1:30 Examples
  • 2:44 Proof
  • 4:14 Lesson Summary
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Lesson Transcript
Instructor: David Karsner
If you wanted to know the sum of the cubes of the first n natural numbers, you could find all the cubes and then add them all together. After the first few cubes, the numbers get large and time- consuming. There are shortcuts to finding this sum.

Sum of the First n Natural Numbers

On many occasions in mathematics, you'll probably catch yourself detecting patterns. These patterns, when applied to a repetitive process, can allow you to complete the computations quickly. Finding the sum of the cubes of the first n natural numbers is one of those occasions. This lesson will show the pattern that will allow you to find the sum of the first n natural numbers quickly. It'll also give a brief proof that the pattern does work in all cases.

If you've been around algebra for any length of time, you've probably heard the phrase Sum of Cubes when referring to the factoring of two cubed terms that have been added together. x3 + 27 would be an example of this kind of sum of cubes. That is not what this lesson is about. This lesson is about finding the sum of the cubes of the first n natural numbers. If you wanted to find the sum of the first three (n = 3) cubes, it would be (1)3 + 23 + 33 or 1 + 8 + 27 = 36. Since n was only 3, it didn't take very long to figure out that the sum of the first three cubes was 36. When n gets larger it becomes a lot more time consuming to find the sum. There's a formula that allows for the sum of the first n cubes to be tallied without a lot of time or trouble. As you can see, the formula is read as:

The sum of cubes for n natural numbers
formulacubes

Examples

Example 1

Let's start with n = 3, since we already know the answer.

(3(3+1)/2)2

((3 x 4)/2)2

(12/2)2

(6)2

Therefore, our answer is 36.

Example 2

Find the sum of the first 12 cubes.

n = 12

n + 1 = 13

12 x 13 = 156

156 / 2 = 78

782 = 6,084

That's a lot quicker than finding the 12 cubes and then adding them together, and you get the same answer as doing it longhand.

There's another means of finding the sum of the cubes of the first n natural numbers. You can sum up the first n numbers and then square your answer. This may sound simpler at first, but summing up those first n numbers can be very time-consuming. If you let n=12, you would need to add 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 = 78 and then square it; 782 = 6,084. The other way is quicker.

Proof

Induction can be used to prove that the sum of the first n natural numbers is the square of:

((n x (n+1)) / 2)2

Step One

The first step of induction is to prove that when n = 1, it'll work. So n = 1, n + 1 = 2. (1)(2)/(2) = 1. The square of 1 = 1. 13 = 1. It works for n = 1.

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