MF51: More patterns with algebra

The patterns formed by triangular numbers and square numbers have generalizations in different directions. One is to three dimensional tetrahedral numbers, and three dimensional pyramidal numbers. Another is to pentagonal numbers, which have a number of interesting features.

This video belongs to Wildberger's MathFoundations series, which attempts to create a coherent and logical framework for modern mathematics.

Category:

Science & Technology

Tags:

• Mathematics
• foundations
• Wildberger
• number
• patterns
• algebra
• triangular
• square
• pentagonal
• numbers
• tetrahedral
• pyramidal

[EclecticSceptic]

I think it's really interesting that the triangular numbers are in the form (n + 1) C 2, and tetrahedral numbers are in the form (n + 2) C 3. In fact, the natural numbers are in the form n C 1. So, all these series can be defined as (n + r) C (r + 1), where r is any non negative integer!

[samruby82]

@samruby82 @samruby82 nevermind, found a good one here: blog.jgc.org/2008/01/proof-tha­t-sum-of-squares-of-first-n.ht­ml

[samruby82]

@jbannon200 thanks for the reply, yes induction is pretty easy for this but I was more interested in a derivation, i was able to build a derivation geometrically but not purely algebraically. I'm more interested in finding out where the equations come from and what they relate to instead of simply proving that they just work.

[samruby82]

Do you ever end up proving the formula for square pyramidal numbers?

(n*(n-1)*(2n+1))/6.

Also, are you planning on showing the inductive proof, I'd be more interested in a derivation.

[mrsheetballz]

the first table is just pascals triangle