Question

Is there a simple way to write these as a kind of formula?

Answer 1

\[\large (a+b+c+...)^2 - (a^2+b^2+c^2 +...)= 2( ab+bc + ac + ... )\]

Answer 2

I want to specifically look at the difference of the square of a sum and the sum of squares, so an equation that looks like this: \[\Large f(g(x))-g(f(x))\] if that makes sense.

Answer 3

If it helps, I'm trying to think of it as the "commutator" of the sum and square functions. But I guess they're not exactly functions? But the reason this seems useful and interesting is that it sort of produces a bunch of terms added together that are kind of like interesting to permutations, combinatorics, and determinants so there might be some nice way of evaluating them? Just a random idea or motivation to anyone else out there.

Answer 4

Your example's a variation of the multinomial theorem, if that helps... which is indeed useful in combinatorics.

Answer 5

I've never heard of the multinomial theorem but I imagine it's sort of like higher order version of the binomial theorem?

Answer 6

Sort of, yeah. It deals with the general form of the expansion of any polynomial with more than two terms, like \((a+b+c)^2\).

Here's the form in general: http://en.wikipedia.org/wiki/Multinomial_theorem#Theorem

Answer 7

Oh ok I've come up with a formula like his before, where the top factorial has the total number of terms and the bottom has the number of factorials with their values inside them adding up to the total inside the factorial on top.

Answer 8

Right, the multinomial coefficient shows up quite frequently when you have a problem like finding the number of ways you can rearrange a string of letters with repeated symbols.

Answer 9

And I think it ends up adding up to like 2^n for binomial, 3^n for trinomials if you add up all the terms I'd guess. Hmmm I don't really know where to take it from here though.

Actually originally I was trying to prove Goldbach's Conjecture that every even number is the sum of two primes since (a+b)^2 - (a^2+b^2)=2ab is always an even number as well and thought I could find a way to bootstrap my way into it that way. Hmmm. Sorry I'm not really going in any specific direction here just sort of throwing up my thoughts or ideas, I like the monomial theorem suggestion though, it is helping. =D

Answer 10

I like to think a lot of the great results of mathematics are a direct result of scattered thoughts. Best of luck :)

Answer 11

Hahaha I can only hope you're right and hope that scattered thoughts don't prevent me from getting a stable job to feed and house myself. XD

Answer 12

Yeah those topics tend to be left out of the curriculum for some reason... lol