Adding a sequence of numbers

kdouglas

ok, I remember being stupidly bored one day and started adding numbers together in my head, i must have been trying to sleep or something, but i noticed a quick way to get the sum of all numbers from n to 0,

i.e. 10 + 9 + 8 ... + 1 = 55

simply ((10+1) / 2) * 10 or ((n + 1) / 2) * n


as far as i can tell, this holds true for all values for n?
i've never actually seen this anywhere, but just wondering is there a name for this method and am i right in saying it works for all values of n?

Michael Collins

Yep, you are indeed correct that it works for all non-negative n. This is a pretty standard series, the sum of the natural numbers in sequence. What you have is the "Sum to n-terms" of that series.

The proof of this is on the Leaving Cert course and uses the method of induction, which is farily straightfoward.

LeixlipRed

This problem was famously given to the mathematician Gauss when he was a child in Germany. His teacher had to leave the class and before he did he set the class the problem of adding the numbers 1 to 100 thinking it would keep them busy for atleast a few minutes. Before he'd left the room Gauss had the answer using the exact same method as above. Pity you're only a couple of hundred years too late

kdouglas

well i knew someone must have thought of it at some stage

Cheers lads

LeixlipRed

Well if you'd never read anything about it and haven't studied much maths then I suppose it's mostly an original thought. Now you just need to build the time machine

kdouglas

well, i only really studied leaving cert maths and a bit in college (computer science), but i knew it wasnt an original idea, someone somewhere must have thought of it at some stage, just wanted to know if anyone was famous for having thought of it or if it was just a well known thing

LeixlipRed

Well Gauss is the one the legend is attached too. Whether it's a true story or not is debatable. If you seen some linear algebra in college you might have come across Gaussian elimination which was named after the same guy

aequinoctium

as well as Gaussian surfaces....

ZorbaTehZ

LeixlipRed wrote:

This problem was famously given to the mathematician Gauss when he was a child in Germany. His teacher had to leave the class and before he did he set the class the problem of adding the numbers 1 to 100 thinking it would keep them busy for atleast a few minutes. Before he'd left the room Gauss had the answer using the exact same method as above. Pity you're only a couple of hundred years too late

The way Gauss looked at it though, was that he noticed 1+99, 2+98, 3+97 etc all summed to 100, and then you need only include the 50 and the other 100. He didn't use that formula to find the answer per se.

LeixlipRed

Of course he didn't use that formula but the thinking behind it was the same. And its actually 1 + 100 = 101, 2+99 = 101, .....

And 50 of those gives you 5050

ZorbaTehZ

No one mentioned it in the thread and since thats the cornerstone of the story itself, and was what displayed his genius at such a young age, the fact that he noticed how it could be simplified, warrants inclusion no?

dan719

But was it not also true that Gauss looked at his a list of numbers belonging to his father and noticed an adding mistake while he was only a toddler?!!

LeixlipRed

dan719 wrote:

But was it not also true that Gauss looked at his a list of numbers belonging to his father and noticed an adding mistake while he was only a toddler?!!

Gauss lived in the late 1700/early 1800s so any "story" about him is unlikely to be 100% accurate. More likely they were anecdotes created long after he lived to add to the legend. Of course they could be true but I've never read anything which indicated they're firm historic fact. Gauss was a genius alright so worrying about the little details is a bit pointless

Son Goku

How about the fact that he discovered quaternions half a century before Hamilton. Or that he discovered Riemannian geometry before Riemann. Giving probability an analytic basis and furthering the understanding of calculus over the complex numbers.

Plus long before Einstein he had his suspicion that space was curved.

LeixlipRed

Son Goku wrote:

How about the fact that he discovered quaternions half a century before Hamilton.

But Hamilton won the race to scribble about quaternians on Broombridge

aequinoctium

But Hamilton won the race to scribble about quaternians on Broombridge

nice

Yakuza

Quick Proof:
Σ(n)= 1 + 2 + 3 +....+(n-2) + (n-1) + n
Write it the other way
Σ(n) = n + (n-1) + (n-2) + ....+3 + 2 + 1

Add the two expressions:

2 * Σ(n) = (n+1) + (n+1) + (n+1)+...+(n+1) + (n+1) + (n+1) = n * (n+1)

So:
Σ(n) = n(n+1) / 2

works for all integers N

Mellor

I highly doubt Guass was the first to simplify it. I've often, as had most people who are good at maths, added the first and last (or second last number) in a series when added to make it a simplier sum, i'm sure it was done before Guass, it just never warrented fame.