# A New Proof Of 1+2+3+...+n = $n(n+1)/2$

Let $f(n)=1+2+3+...+n$.

Now, we have $f(n)-f(n-1)=n$, and we will prove $f(n)+f(n-1)=n^2$ by induction.

Since the proposition is true for $n=1$, let us assume it is true for $n=k$. Now for $n=k+1$, we have: $f(k)+f(k+1)=[f(k)+f(k-1)]+2k+1=k^2+2k+1=(k+1)^2$. So, we conclude that $f(n)+f(n-1)=n^2$ for n belongs to N. Now, we have a system of simultaneous equations: $f(n)+f(n-1)=n^2$ $f(n)-f(n-1)=n$ solving them, we have $2f(n)=n^2+n$ which yields: $f(n)= \boxed {n(n+1)/2}$

Note by Mohammed Imran
1 month ago

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I guess you meant f(n)-f(n-1)=n (not 2) at two instances.

- 1 month ago

thank you very much!!!

- 1 month ago

Cool

- 2 weeks ago

thank you very much

- 2 weeks ago

How is this proof?

- 2 weeks ago

Well I guess that you used the mathematical induction in this problem, right? It's a kind of simple method to prove such these kind of mathematical statements. I like it.

- 2 days, 20 hours ago

Thx!!!

- 2 days, 8 hours ago

Oh nice, where did you see this?

- 3 weeks, 6 days ago

this is the original proof of the theorem by gauss

- 3 weeks, 6 days ago

Then why did you type it as yours?🤔

- 3 weeks, 6 days ago

I said chew seong's proof is not original

- 3 weeks, 6 days ago

Ok

- 3 weeks, 6 days ago

Finally, note that if all the relations are freelance of the flexible, for example, if there's no I within the principle but the adjustable beneath the letter of the alphabet is I, then all the positions are relentless. The number of terms is successful to be given by the commencement and ending values. Here, all the relations are secure (constant) at lawessayteacher.co.uk

- 3 weeks, 4 days ago

Hi Can you please not use this kind of vocabulary when doing math. I cannot understand a thing. And it is very annoying to keep checking the meaning of each word. And I believe that there is nothing related to math here. So, please retype this paragraph and send it properly

- 3 weeks, 4 days ago

Even i dont understand a thing. I believe there is a grammar mistake

- 3 weeks, 3 days ago