Tetrahedral Numbers: Sum of 'n' Consecutive Triangular Numbers: Proof

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In this video I go over tetrahedral numbers and provide a proof of their formula using mathematical induction. This video was requested by Dominic from Strode College in the UK, and is a good follow-up from my earlier video on the sum of consecutive integers.

A tetrahedral number is a number that can be visually represented as 3 sided pyramid, that is a tetrahedron. For example the numbers 1, 4, and 10 can all be visualized as a stack of balls in a tetrahedron pattern. In fact, the tetrahedron is just the sum of consecutive "triangular numbers", which are numbers that can be represented by an equilateral triangle.

I cover triangular or triangle numbers as well, and show that their formula is the sum of n consecutive integers. Finally, I use the triangle number formula as well as the principle of mathematical induction to prove the formula for a tetrahedron number. This was the first time I had learned about tetrahedron numbers so this was an interesting topic to cover. If you have any other video requests please let me know and I may cover it!

The topics covered as well as their timestamps are listed below.

Introduction: 0:00
Topics to Cover: 0:45

Shoutout to Dominic from Strode College: 1:22
Tetrahedral Numbers: 2:26
Triangular Numbers: 7:56
- Formula for a Triangular Number: 8:42
- Visualizing the Triangular Number Formula: 10:33
Formula for a Tetrahedral Number: 13:51
- Proof by Mathematical Induction: 17:37

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Tetrahedral Numbers: Sum of 'n' Consecutive Triangular Numbers: Proof

Tetrahedral Numbers.jpeg

Topics to Cover

The timestamps will be included in the video description for each topic listed below.

Shoutout to Dominic from Strode College
Tetrahedral Numbers
Triangular Numbers
- Formula for a Triangular Number
- Visualizing the Triangular Number Formula
Formula for a Tetrahedral Number
- Proof by Mathematical Induction

Shoutout to Dominic from Strode College!

Note that this video was requested by Dominic from Strode College in the UK: https://www.strode-college.ac.uk/

If you have a video request feel free to comment below or email me at [email protected] and I may make the video!

Tetrahedral Numbers

A tetrahedral number, or triangular pyramidal number, is a number that can be geometrically visualized as a triangular pyramid with three sides, which is called a tetrahedron.

A number is considered a tetrahedral number if we can stack an equivalent number of balls into a tetrahedron; note that 1 is considered a tetrahedral number too.

The first 3 tetrahedral numbers are 1, 4, and 10.

The next tetrahedral number is 20.

Note that each consecutive tetrahedral number simply adds a bigger triangular base to the sum.

In fact, as per Dominic's email, a tetrahedral number is just the sum of triangular numbers.

Triangular Numbers

A triangular number or triangle number is a number that can be geometrically visualized as an equilateral triangle.

The first 6 triangle numbers are 1, 3, 6, 10, 15, and 21; and are shown below.

Formula for a Triangular Number

The formula for a triangular number can be seen above as simply the sum of n integers; notice how in each triangle number we add the n-th number to the previous triangle number.

In other words, a triangular number is just the sum of n consecutive integers, which I proved in my earlier video.

Visualizing the Triangular Number Formula

Note that we can visually confirm the formula for a triangular number, by drawing a n x (n + 1) rectangle and dividing by 2, such as in the example below:

Formula for a Tetrahedral Number

The formula for a tetrahedral number is just the sum of the first n triangular numbers, and which can be evaluated as:

Proof by Mathematical Induction

Recall the principle of mathematical induction from my earlier video:

@mes/problems-plus-example-4-mathematical-induction
Retrieved: 23 August 2022
Archive: https://archive.ph/wip/gHATr

In our case, let's start with the base case (n = 1), which we can verify is true by comparing the Tetrahedral formula with the sum of the first triangular number.