Solving Series Questions: 1+2+3...+n & 1^2+2^2+3^2...+n^2+n^3

[HALO3]

If
1) 1+2+3...+n =n(n+1)/2

Then what is
2) $$1^2+2^2+3^2+...+n^2= ?$$
3) $$1^3+2^3+3^3+...+n^3= ?$$

If someone can explain me how the first one came and how can i proceed to solve the second and third, it would be really grateful.

 

[HallsofIvy]

The first one is famous! When Gauss was just a child, the story goes, his elementary school teacher set the entire class the problem of adding all numbers from 1 to 100, just to keep them busy. Gauss wrote a single number on his paper, and then sat staring at the teacher. What he had done was recognise that is he wrote out the sum (which he did "in his head",
1+ 2+ 3+ ...+ 100 and then reversed it
100+99+ 97+...+1, each (vertical) pair will add to 101. Since there are 100 pairs, they all add to 100*101= 10100. But we added 1 to 100 twice: the correct sum is half of that: 10100/2= 5050. Now try it with numbers up to n:
1+ 2+ 3+ ...+ n
n+n-1+n-2+...+1

Now each vertical pair adds to n+1 and there are n pairs: total sum, n(n+1). Because we have added from 1 to n twice there, the correct sum is n(n+1)/2.

(There are, now, many different ways you could do that sum.)

A much harder calculation shows that the sum of squares is equal to n(n+1)(2n+1)/6.

Oddly enough the sum of cubes is equal to (n(n+1)/2)².

 

[CompuChip]

Usually 2) is used as an example to illustrate the idea of proof by induction. If you want I can explain that, but I don't know if you're already supposed to know this?

 

[nanoWatt]

I actually had found that formula on my own before I realized it was already a famous formula. I called it: a number, times one half the number, plus one half the number. I think I was about 10 or 12 when I thought I had discovered something I termed the "preconsecutive".

 

[Differential]

i am sorry my english is not well .
The first one: n+1=(n-1)+2=(n-2)+3=...
if n is double that the first one : (n+1)*n/2
if n is not double that the first one : [(n+1)*(n-1)/2]+(n+1)/2

 

[CompuChip]

Welcome to PF, Differential.
Your English is not a problem (math is universal, right?) but I don't really understand what you're trying to do? Is it supposed to be a proof of the original question?

 

[robert Ihnot]

Hallof Ivy: The first one is famous! When Gauss was just a child, the story goes, his elementary school teacher set the entire class the problem of adding all numbers from 1 to 100, just to keep them busy.

Please, excuse me for throwing a little cold water here. When I first heard the story in school, I was under the impression that this "Gaussian Sum," was actually discovered first
by young Gauss. Very inspiring! However, if that were true, how could the teacher have found the sum?

In any case, the sum was known to Pythagoras.

Now http://www.americanscientist.org/template/AssetDetail/assetid/50686?&print=yes points out a certain amount of questions here. The story was published by Sartonius in 1856, but he gave no amounts involved. It has been argued that the sum from 1-100 has only been generally settled on in recent times, and others amounts have been supposed, such as 1-50, 1-80, or even 1-1000. Depending upon(?) what some feel schoolchildren are capable of.

I remember reading something quite different in Men of Mathematics by Eric T Bell, and he proposes that the question was sum up the series: 81297+81495+81693+++100899, where each difference is 198.

Of course, the idea of Bell is many times harder to work with than a simple sum like 1-100, and it all suggests that, as some have said, The story has grown with the telling. A few have even suggested that the event never even occurred.

However, to its credit, it’s a GOOD STORY and has been an inspiration to many a budding math student. Doubtless it will live on and on. Perhaps, growing even more wonderful as time passes.

 

[VietDao29]

Hallof Ivy: The first one is famous! When Gauss was just a child, the story goes, his elementary school teacher set the entire class the problem of adding all numbers from 1 to 100, just to keep them busy.

Please, excuse me for throwing a little cold water here. When I first heard the story in school, I was under the impression that this "Gaussian Sum," was actually discovered first
by young Gauss. Very inspiring! However, if that were true, how could the teacher have found the sum?

I'm not sure if this is true, but some book says that: The teacher, himself, didn't even know the result, or the way to calculate it. He just gave it to the class as the punishment (for not being quiet enough, or something along the lines). Some book even says that the teacher was tired, and just threw out a random problem, he could think of for the class to solve, just to keep them busy for a while. And, Gauss was the only student in the class who could solve it. Well, it's just a story, and it's long ago. So, there are many versions, all of which differ each other a bit. Whether the story is true or not, there is a truth we cannot deny, i.e Gauss is one of the most talented mathematicians who've ever lived.

 

[HallsofIvy]

Hallof Ivy: The first one is famous! When Gauss was just a child, the story goes, his elementary school teacher set the entire class the problem of adding all numbers from 1 to 100, just to keep them busy.

Please, excuse me for throwing a little cold water here. When I first heard the story in school, I was under the impression that this "Gaussian Sum," was actually discovered first
by young Gauss. Very inspiring! However, if that were true, how could the teacher have found the sum?

In any case, the sum was known to Pythagoras.

Now http://www.americanscientist.org/template/AssetDetail/assetid/50686?&print=yes points out a certain amount of questions here. The story was published by Sartonius in 1856, but he gave no amounts involved. It has been argued that the sum from 1-100 has only been generally settled on in recent times, and others amounts have been supposed, such as 1-50, 1-80, or even 1-1000. Depending upon(?) what some feel schoolchildren are capable of.

I remember reading something quite different in Men of Mathematics by Eric T Bell, and he proposes that the question was sum up the series: 81297+81495+81693+++100899, where each difference is 198.

Of course, the idea of Bell is many times harder to work with than a simple sum like 1-100, and it all suggests that, as some have said, The story has grown with the telling. A few have even suggested that the event never even occurred.

However, to its credit, it’s a GOOD STORY and has been an inspiration to many a budding math student. Doubtless it will live on and on. Perhaps, growing even more wonderful as time passes.

First, I didn't say Gauss was the first to "discover" that formula, only that he was able to find and use it when he was very young. Second, I don't see why you think the teacher would have to know that formula in order to find the sum himself. He could easily have just done the sum, 1+ 2+ 3+ ...+ 99+ 100, directly, before the class. Or he simply didn't care what the sum was! In any case, what I said was "The story goes ...".

 

[robert Ihnot]

Hallsof Ivy: In any case, what I said was "The story goes ...".

I realized that you did say, "The story goes," I just quoted you because you went over a familiar story. In fact, I like the story myself.

It wasn't intended as a personal attack or any such thing.