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

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

• Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
• Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
• Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
• Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$ ... $$ or $ ... $ to ensure proper formatting.
2 \times 3 $2 \times 3$
2^{34} $2^{34}$
a_{i-1} $a_{i-1}$
\frac{2}{3} $\frac{2}{3}$
\sqrt{2} $\sqrt{2}$
\sum_{i=1}^3 $\sum_{i=1}^3$
\sin \theta $\sin \theta$
\boxed{123} $\boxed{123}$

Sort by:

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