Math Terminology Question

[hummingwolf]

Now that half of you have been scared away by the subject line:

Okay, numbers which are the sums of the natural numbers from 1 through [whatever] are called triangular numbers. You can see why if you do a little dot drawing of, say, the sum of the numbers from 1 through 4:

*
**
***
****

The above shows you that 10 is a nice triangular number, which you already knew if you've ever been bowling. Other triangular numbers include 36 (the sum of the numbers from 1 through 8) and 666 (sum of numbers from 1 through 36).

Here's my question: Is there a general name for the numbers made up of sums of evenly-spaced numbers? Like the sum of all the odd numbers between 15 and 71, or the sums of the numbers in this series:
-23
-12
-1
10
21
32
43
54
65
etc.?

I've long known how to figure out what the values of the numbers are, but have no idea what to call them.

Comments

[cowboybud.livejournal.com]

Math is hard.

[hummingwolf.livejournal.com]

Heh. Language is harder, but most of us have more practice using words than numbers.

[deborahlane.livejournal.com]

I don't know, but I have a friend who might...I've sent him your way! :)

[hummingwolf.livejournal.com]

Okay, thank you!

[stronae.livejournal.com]

I'm not sure if there's a name, but the formula is simple. If the 'base' of the sum is b, and there are n terms, and the skip between each term is k, then the total sum is

bn + ((n(n-1))/2)k.

In your example above, straight addition gives 189.
b = -23, k = 11, and n = 9, and plugging these in the above formula also gives 189. Notice that the (n-1)^st triangular sum appears as ((n(n-1))/2). :)

[hummingwolf.livejournal.com]

Right, the formula wasn't what I wanted. I just wanted some idea of what to call them. :-)

The way I actually learned to figure them was, if I knew I wanted to add those numbers up to some number n (not the number of terms, but, e.g., number 65 on that list), I could use

((n+b)(n+k-b)) / 2k

And the sum of the sums is
((n+2b)(n+k-b)(n+2k-b)) / 2*3k²

And the sum of the sums of sums is something you can surely figure out so I won't bother typing up any more. :-)

[moomlyn.livejournal.com]

I don't know, sorry.