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

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

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

Download Video Notes: https://1drv.ms/b/s!As32ynv0LoaIibBbOJ0HyyI1sOoEhw?e=VxJ3Yv

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Related Videos:

Summations Playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0HyAqJRuDIRPUAvO_3ytXtu

Sum of the squares of "n" Consecutive integers - Simple Proof: https://peakd.com/hive-128780/@mes/sum-of-the-squares-of-n-consecutive-integers-proof .

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