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

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

Now, we have f(n)f(n1)=nf(n)-f(n-1)=n, and we will prove f(n)+f(n1)=n2f(n)+f(n-1)=n^2 by induction.

Since the proposition is true for n=1n=1, let us assume it is true for n=kn=k. Now for n=k+1n=k+1, we have: f(k)+f(k+1)=[f(k)+f(k1)]+2k+1=k2+2k+1=(k+1)2f(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(n1)=n2f(n)+f(n-1)=n^2 for n belongs to N. Now, we have a system of simultaneous equations: f(n)+f(n1)=n2f(n)+f(n-1)=n^2 f(n)f(n1)=nf(n)-f(n-1)=n solving them, we have 2f(n)=n2+n2f(n)=n^2+n which yields: f(n)=n(n+1)/2f(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.

K T - 1 month ago

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thank you very much!!!

Mohammed Imran - 1 month ago

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Cool

Nabeel Akhter - 2 weeks ago

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thank you very much

Mohammed Imran - 2 weeks ago

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How is this proof?

Mohammed Imran - 2 weeks ago

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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.

Anh Khoa Nguyễn Ngọc - 2 days, 20 hours ago

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Thx!!!

Mohammed Imran - 2 days, 8 hours ago

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Oh nice, where did you see this?

Nitin Kumar - 3 weeks, 6 days ago

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this is the original proof of the theorem by gauss

Mohammed Imran - 3 weeks, 6 days ago

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Then why did you type it as yours?🤔

Nitin Kumar - 3 weeks, 6 days ago

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@Nitin Kumar I said chew seong's proof is not original

Mohammed Imran - 3 weeks, 6 days ago

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@Mohammed Imran Ok

Nitin Kumar - 3 weeks, 6 days ago

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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

amelie jack - 3 weeks, 4 days ago

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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

Mohammed Imran - 3 weeks, 4 days ago

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

Nitin Kumar - 3 weeks, 3 days ago

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