What is the formula for finding the nth partial sum?

[mark2142]

Homework Statement

"When the famous mathematician C. F. Gauss (see page 290) was a schoolboy, his teacher posed this problem (To find sum 1+2+3+...+100) to the class and expected that it would keep the students busy for a long time. But Gauss answered the question almost immediately. His idea was this:
Since we are adding numbers produced according to a fixed pattern, there must also be a pattern (or formula) for finding the sum."

Relevant Equations

(From the book Precalculus by J. Stewart. Chapter 12 Sequence and series)

Since we are adding numbers produced according to a fixed pattern, there must also be
a pattern (or formula) for finding the sum.

Hi, We use this method to find the $S_n$. I don't understand how the sum will also be in a pattern. Can someone please explain this line in bold?

 

[FactChecker]

I wouldn't say that the statement in question was a "method". In fact, it doesn't even hint at what the method might be. I would call it a "motivation" that there must be some methodical method. Gauss would see that method easily because he was a genius beyond our imagination.
Do you know the answer and how to derive it? For homework problems, if you show us your work then we can give some hints and guidance.

 

[mark2142]

I wouldn't say that the statement in question was a "method". In fact, it doesn't even hint at what the method might be. I would call it a "motivation" that there must be some methodical method. Gauss would see that method easily because he was a genius beyond our imagination.
Do you know the answer and how to derive it? For homework problems, if you show us your work then we can give some hints and guidance.

The method is given in the book. Gauss wrote numbers in increasing order and then below it in decreasing order and summed. This can be used to find not just sum of 100 terms but also nth terms. The method itself is not the kind that we can call him genius yet. It’s just a trick.

There must be something that made him believe that sum should also be in pattern. What is it? How a simple series of numbers in pattern can have sum in pattern?

 

[PeroK]

There must be something that made him believe that sum should also be in pattern. What is it? How a simple series of numbers in pattern can have sum in pattern?

That's the whole idea of mathematics, isn't it? As opposed to arithmetic, which we might take to mean specific one-off calculations.

 

[mark2142]

I think I see it. $S1=1$, $S2=3$, $S3=6$. There is a pattern. That is $S_n=S_{n-1} + n$. Yes?
(I am a genius too!)

 

[FactChecker]

I think I see it. $S1=1$, $S2=3$, $S3=6$. There is a pattern. That is $S_n=S_{n-1} + n$. Yes?
(I am a genius too!)

Of course, this was trivial for Gauss. If you are making fun of his ability, I suggest that you pick another target. :-)

 

[FactChecker]

I think I see it. $S1=1$, $S2=3$, $S3=6$. There is a pattern. That is $S_n=S_{n-1} + n$. Yes?
(I am a genius too!)

IMO, no. This is self-evident and gets you nowhere.

 

[mark2142]

Of course, this was trivial for Gauss. If you are making fun of his ability, I suggest that you pick another target. :-)

I am not making fun of anybody. I was just asking how can one reach to a believe. There must be something that made him believe that. Genius is a very heavy word. If he was a proven genius I am an unproven one.

gets you nowhere.

If you can't answer my question then I think this is enough for me to believe that sum should also be in pattern.

 

[FactChecker]

I am not making fun of anybody. I was just asking how can one reach to a believe. There must be something that made him believe that. Genius is a very heavy word. If he was a proven genius I am an unproven one.

If you think that Gauss' fame is due to some trivial trick that he saw as a child, then you are wrong.

If you can't answer my question then I think this is enough for me to believe that sum should also be in pattern.

But IMHO $S_n = S_{n-1} + n$ is a simple restatement of the problem and gets you no closer to the answer. That is not the simple trick that the book should be talking about.

 

[mark2142]

It does not get me closer to the answer but at least it shows sum is indeed in pattern when terms are.
I think I should ask Gauss himself.

 

[fresh_42]

Homework Statement:: "When the famous mathematician C. F. Gauss (see page 290) was a schoolboy, his teacher posed this problem (To find sum 1+2+3+...+100) to the class and expected that it would keep the students busy for a long time. But Gauss answered the question almost immediately. His idea was this:
Since we are adding numbers produced according to a fixed pattern, there must also be a pattern (or formula) for finding the sum."
Relevant Equations:: (From the book Precalculus by J. Stewart. Chapter 12 Sequence and series)

Can someone please explain this line in bold?

The used pattern was adding $1$ to $100$, $2$ to $99$, etc. leaving the multiplication $50\cdot 101 = \dfrac{n}{2}(n+1)$ which is a pattern that uses a) the given pattern (number +one), b) produces a new pattern (summing equal terms), that c) immediately produced the formula.

 

[PeroK]

It does not get me closer to the answer but at least it shows sum is indeed in pattern when terms are.

I'm not sure what sort of "answer" you are looking for. I suspect that the sum of a simple arithmetic sequence is something that countless talented mathematics students have worked out for themselves before being taught it. It's a nice story, and perhaps Gauss was exceptionally young when he worked it out, but I don't think it's something that takes genius to work out for yourself.

It's a bit like being a music student and being mystified that any human being ever wrote a song!

 

[fresh_42]

It's a bit like being a music student and being mystified that any human being ever wrote a song!

The genius part doesn't lie in the solution. The genius part, especially for an IIRC 6-year-old was to think about a solution instead of starting to run! It opened him the path, i.e. provided the necessary opportunities to become what he became.

 

[FactChecker]

It does not get me closer to the answer but at least it shows sum is indeed in pattern when terms are.
I think I should ask Gauss himself.

I think that $S_{n+1} = S_n + (n+1)$ actually makes it harder. It is harder (for me at least) to see that the order of the terms can be reversed and added to the original terms to give $n(n+1)/2$.

I would interpret $1+2+...+n$, itself, to be the pattern that the book is referring to. There are several other summations where we look to reverse the order of the terms to get a simple formula for the sum.

 

[mark2142]

But IMHO is a simple restatement of the problem and gets you no closer to the answer. That is not the simple trick that the book should be talking about.

I have attached a part from the book where they explained how the formula was derived.

 

[FactChecker]

I have attached a part from the book where they explained how the formula was derived.

Right. In the form $1+2+...+n$, the formula is easy to derive. I think that putting it in the form $S_{n+1}=S_n+(n+1)$ makes the derivation harder to see. IMO, it is a step in the wrong direction.