# Odd thing I noticed about a simple mathematical pattern **URL:** \ **Category:** Factual Questions\ **Created:** [October 15, 2017, 6:46pm UTC](https://boards.straightdope.com/t/odd-thing-i-noticed-about-a-simple-mathematical-pattern/798967 "2017-10-15T18:46:55Z")\ **Posts on this page:** 12\ **Page:** 1
**Author:** ![Left\_Hand\_of\_Dorkness](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/left_hand_of_dorkness/32/7156_2.png) [@Left\_Hand\_of\_Dorkness](https://boards.straightdope.com/u/Left_Hand_of_Dorkness)\ **Post date:** [October 15, 2017, 6:46pm UTC](https://boards.straightdope.com/t/odd-thing-i-noticed-about-a-simple-mathematical-pattern/798967/1 "2017-10-15T18:46:55Z")
I was carrying my sleepy cranky four-year-old up a hiking trail yesterday, which gave my brain time to wander, and I noticed something. Take the common pattern made with adding things to a triangle, where you create a new row by increasing the number of things in the row by 1 each time, like so: X XX XXX XXXX XXXXX XXXXXX XXXXXXX XXXXXXXX and so on. As you go down the triangle, your running total looks like this: 1 3 6 10 15 21 28 36 and so on. With me so far? So I noticed that between the first and second row, your running total is multiplied by 3 (1 x 3 = 3). Between the second and third rows, your running total is multiplied by 2 (3 x 2 = 6). Between the third and fourth it got tricky. But here’s what I noticed: 1 x 3/1 = 3 2 x 4/2 = 6 6 x 5/3 = 10 10 x 6/4 = 15 15 x 7/5 = 21 21 x 8/6 = 28 28 x 9/7 = 36 And, presumably, so on. In other words, if you add one to the numerator and denominator of the previous fraction and multiply your current running total by the resulting number, you’ll get the next running total. This blew my mind, because at first glance, the two patterns ({1, 3, 6, 10, 15} and {3/1, 4/2, 5/3, 6/4, 7/5}) are completely unrelated–and the second pattern isn’t one I’ve ever seen elsewhere. But surely there’s a reason for it. I’m decent at understanding math, but my vocabulary is pretty limited. Can anyone explain in very small words why this pattern works? ---
**Author:** ![kk\_fusion](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/kk_fusion/32/15276_2.png) [@kk\_fusion](https://boards.straightdope.com/u/kk_fusion)\ **Post date:** [October 15, 2017, 7:08pm UTC](https://boards.straightdope.com/t/odd-thing-i-noticed-about-a-simple-mathematical-pattern/798967/2 "2017-10-15T19:08:13Z")
All numerators except the last two cancel out with all denominators except the first 2, so you’ll end up with n \* (n + 1) / 2. ---
**Author:** ![Left\_Hand\_of\_Dorkness](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/left_hand_of_dorkness/32/7156_2.png) [@Left\_Hand\_of\_Dorkness](https://boards.straightdope.com/u/Left_Hand_of_Dorkness)\ **Post date:** [October 15, 2017, 7:10pm UTC](https://boards.straightdope.com/t/odd-thing-i-noticed-about-a-simple-mathematical-pattern/798967/3 "2017-10-15T19:10:35Z")
> [@kk\_fusion](#): > > All numerators except the last two cancel out with all denominators except the first 2, so you’ll end up with n \* (n + 1) / 2. I’m afraid I don’t follow. Do you mind using littler words, or making explicit the things you’re thinking are obvious? ---
**Author:** ![kk\_fusion](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/kk_fusion/32/15276_2.png) [@kk\_fusion](https://boards.straightdope.com/u/kk_fusion)\ **Post date:** [October 15, 2017, 7:17pm UTC](https://boards.straightdope.com/t/odd-thing-i-noticed-about-a-simple-mathematical-pattern/798967/4 "2017-10-15T19:17:36Z")
The Gaussian sum of the first numbers from 1 through n is n \* (n + 1) / 2. Now if you derive the n-th term of the series by starting at 1 and multiplying all the ratios 3/1 \* 4/2 \* 5/3 \* … \* n/(n-2) \* (n+1)/(n-1), you’ll find most numbers cancelling out and also leaving n \* (n + 1) / 2. ---
**Author:** ![Trinopus](https://avatars.discourse-cdn.com/v4/letter/t/2bfe46/32.png) [@Trinopus](https://boards.straightdope.com/u/Trinopus)\ **Post date:** [October 15, 2017, 7:41pm UTC](https://boards.straightdope.com/t/odd-thing-i-noticed-about-a-simple-mathematical-pattern/798967/5 "2017-10-15T19:41:19Z")
> [@Left\_Hand\_of\_Dorkness](#): > > . . . I’m decent at understanding math, but my vocabulary is pretty limited. Can anyone explain in very small words why this pattern works? The formula, in general, for the sum of the items in that triangle, is n(n+1)/2 So, for the sum of numbers 1 through, say, 7, you get 7\*8 / 2 = 28 Now, bump up to the next level. The sum will be (n+1)\*(n+2)/2. And the ratio of those two numbers – (n+1)(n+2)/2 / n(n+1)/2 – is (n+2)/n So if 28 is the 7th sum, 28 \* 9/7 = 36, which is the eighth sum. 36 \* 10/8 = 45 which is the ninth sum, and so on. You can specify the items explicitly – n(n+1)/2 – or incrementally – N[n+1] = N[n] \* n+2/n A little algebra shows that each of these two ways to look at the series can be manipulated to reveal the other. Lovely stuff! Gauss, it is said, figured this out by cutting the triangle in half and putting the two parts together as a rectangle. ---
**Author:** ![Left\_Hand\_of\_Dorkness](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/left_hand_of_dorkness/32/7156_2.png) [@Left\_Hand\_of\_Dorkness](https://boards.straightdope.com/u/Left_Hand_of_Dorkness)\ **Post date:** [October 15, 2017, 7:52pm UTC](https://boards.straightdope.com/t/odd-thing-i-noticed-about-a-simple-mathematical-pattern/798967/6 "2017-10-15T19:52:24Z")
> [@Trinopus](#): > > The formula, in general, for the sum of the items in that triangle, is n(n+1)/2 > > So, for the sum of numbers 1 through, say, 7, you get 7\*8 / 2 = 28 > > Now, bump up to the next level. The sum will be (n+1)\*(n+2)/2. > > And the ratio of those two numbers – (n+1)(n+2)/2 / n(n+1)/2 – is (n+2)/n > > So if 28 is the 7th sum, 28 \* 9/7 = 36, which is the eighth sum. 36 \* 10/8 = 45 which is the ninth sum, and so on. > > You can specify the items explicitly – n(n+1)/2 – or incrementally – N[n+1] = N[n] \* n+2/n > > A little algebra shows that each of these two ways to look at the series can be manipulated to reveal the other. Lovely stuff! > > Gauss, it is said, figured this out by cutting the triangle in half and putting the two parts together as a rectangle. Huh. I think I need to play with this a bit to understand it. ---
**Author:** ![DSYoungEsq](https://avatars.discourse-cdn.com/v4/letter/d/c6cbf5/32.png) [@DSYoungEsq](https://boards.straightdope.com/u/DSYoungEsq)\ **Post date:** [October 16, 2017, 3:10pm UTC](https://boards.straightdope.com/t/odd-thing-i-noticed-about-a-simple-mathematical-pattern/798967/7 "2017-10-16T15:10:00Z")
> [@Trinopus](#): > > Gauss, it is said, figured this out by cutting the triangle in half and putting the two parts together as a rectangle. I thought the story was that he figured it out by folding the list of numbers being added back upon itself, and noting that the vertical sums were identical. At the age of seven, or something like that. :eek: ---
**Author:** ![OldGuy](https://avatars.discourse-cdn.com/v4/letter/o/3bc359/32.png) [@OldGuy](https://boards.straightdope.com/u/OldGuy)\ **Post date:** [October 16, 2017, 3:16pm UTC](https://boards.straightdope.com/t/odd-thing-i-noticed-about-a-simple-mathematical-pattern/798967/8 "2017-10-16T15:16:10Z")
I think the simplest way to state this is: The sum of 1 + 2 + …+ n = n(n+1)/2 The sum of 1 + 2 + …+ n + n+1 = (n+1)(n+1+1)/2 The ratio is clearly (n+2)/n which is what you found. ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [October 16, 2017, 4:26pm UTC](https://boards.straightdope.com/t/odd-thing-i-noticed-about-a-simple-mathematical-pattern/798967/9 "2017-10-16T16:26:05Z")
> [@DSYoungEsq](#): > > I thought the story was that he figured it out by folding the list of numbers being added back upon itself, and noting that the vertical sums were identical. At the age of seven, or something like that. :eek: Knowing Gauß more likely he figured out half a dozen different proofs and wrote them down in his notebook. ---
**Author:** ![Hari\_Seldon](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hari_seldon/32/5173_2.png) [@Hari\_Seldon](https://boards.straightdope.com/u/Hari_Seldon)\ **Post date:** [October 16, 2017, 4:39pm UTC](https://boards.straightdope.com/t/odd-thing-i-noticed-about-a-simple-mathematical-pattern/798967/10 "2017-10-16T16:39:54Z")
Patterns of this sort occur with all the terms in the series for sums of powers (that is, in the formula for sums of the first n squares, first n cubes, etc.) I just learned this about a month ago from a lecture by John Conway on Faulhaber’s triangle. It gets kind of technical, but this pattern of multiplying by a fraction gotten by adding one to the numerator and denominator of the preceding fraction persists. But in this case, to go from n(n+1)/2 to (n+1)(n+2)/2 you obviously have to multiply by (n+2)/n. ---
**Author:** ![Buck\_Godot](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/buck_godot/32/6573_2.png) [@Buck\_Godot](https://boards.straightdope.com/u/Buck_Godot)\ **Post date:** [October 16, 2017, 8:45pm UTC](https://boards.straightdope.com/t/odd-thing-i-noticed-about-a-simple-mathematical-pattern/798967/11 "2017-10-16T20:45:39Z")
Oh, and as to why the sum of the first n integers is (n)\*(n+1)/2: Suppose I want to add the numbers 1+2+3+4+5+6+7+8+9+10 I can rewrite this as (1+10) + (2+9) + (3+8) + (4+7) + (5+6) you will note that all of the pairs add to 11, and that there are 5 of them so the total is 55. In general for any even number n, I can do the same thing 1+2+…n= (1+n)+(2+(n-1))+(3+(n-2))+…(n/2+(n/2+1)). All of the pairs add up to (n+1) and there are n/2 of them so the total is (n+1)\*n/2. If n is an odd number, you can just start counting from 0. 0+1+…n=(0+n)+(1+(n-1))+(2+(n-2))+…+((n-1)/2+(n+1)/2). All add up to n and there are (n+1)/2 such pairs so you again get n\*(n+1)/2 as the total. ---
**Author:** ![Aspidistra](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/aspidistra/32/3894_2.png) [@Aspidistra](https://boards.straightdope.com/u/Aspidistra)\ **Post date:** [October 16, 2017, 9:09pm UTC](https://boards.straightdope.com/t/odd-thing-i-noticed-about-a-simple-mathematical-pattern/798967/12 "2017-10-16T21:09:58Z")
I think geometrically is the nicest way to think of this. Let’s call your n’th sum **Triangle(n)** Now, there’s also a shape Rectangle(n) which you can make by duplicating Triangle(n), spinning the second one round and plonking it on top. Lets use 5 as the example: **00000 X0000 XX000 XXX00 XXXX0 XXXXX** (I made the second triangle out of circles not crosses for clarity) At this point it’s really obvious why the formula for **Triangle(n) is n(n+1)/2**. It’s half of **Rectangle(n)** which is **n(n+1)** So each line in the OP’s multiplication sum is something of the form: **Triangle(n) \* {something} = Triangle(n+1)** We could do that with rectangles instead … **Rectangle(n) \* {something} = Rectangle(n+1)** Gonna be the same ‘something’ in both cases, clearly, since we’ve simply multiplied both sides by two. What is the ‘something’? Well, looking at, say,\*\* Rectangle(5)\*\* and **Rectangle(6)** it’s not immediately obvious… **00000 X0000 XX000 XXX00 XXXX0 XXXXX** and **000000 X00000 XX0000 XXX000 XXXX00 XXXXX0 XXXXXX** How _do_ you turn the first total into the second by multiplication? Looks tricky. After all, we’ve \*added \*one both horizontally and vertically - seems like figuring out a \*multiplier \*in the general case wouldn’t be that easy. Oh, unless you _rotate the first rectangle 90 degrees_ \*\*XXXXX0 XXXX00 XXX000 XX0000 X00000 - {something} = 000000 X00000 XX0000 XXX000 XXXX00 XXXXX0 XXXXXX \*\* Now all the lines have the same width and it’s obvious what we do - divide by the height of the first figure (5), multiply by the height of the second (7) **Rectangle(5) \* (7/5) = Rectangle(6)** Or more generally… **Rectangle(n) \* (n+2/n) = Rectangle(n+1)** and therefore of course **Triangle(n) \* (n+2/n) = Triangle(n+1)** And we’re back to the OP…