# Thread: 1 + 2 + 3 + ... + n

1. ## 1 + 2 + 3 + ... + n

I'm reading What is mathematics? and I got confused in page twelve where it says about Arithmetical progression.

I don't understand the "Adding the number ( r + 1)" thing.

This is the page:

2. the image you have posted is not visible probably due to some site problem. If you can tell me in which section of the book you have this then i can help.

3. We'll call this sum S.

We can write S = 1 + 2 + 3 + ... + (n - 2) + (n - 1) + n.

We can also write it as S = n + (n - 1) + (n - 2) + ... + 3 + 2 + 1.

If we add the sums written in these two forms together we get

2S = (n + 1) + (n - 1 + 2) + (n - 2 + 3) + ... + (3 + n - 2) + (2 + n - 1) + (1 + n)

2S = (n + 1) + (n + 1) + (n + 1) + ... + (n + 1) + (n + 1) + (n + 1)

2S = n(n + 1)

S = n(n + 1)/2.

4. I tried to upload it again abhishekkgp.

I'm trying to understand it Prove It.

5. Hello, GustavoB!

I'm reading What is mathematics? and I got confused in page twelve
where it says about Arithmetical progression.

I don't understand the "Adding the number (r + 1)" thing.

It sounds like a Proof by Induction.

Are you familiar with it?

6. Originally Posted by GustavoB

I tried to upload it again abhishekkgp.

I'm trying to understand it Prove It.
i guess soroban's post will solve your query. prove-it has proved it using a different method.

7. The Best Method Is Prove-It 's method

Induction is a second alternative

8. What sounds strange to me is that he adds a (r + 1) to both sides of the equation, then it became r(r + 1) + 2(r+1) and so on.

I don't understand this process.

Prove it already tried to explain me but i still can't see why to do a Sum S.

This page is a part of the book where the author tries to explain Proof by induction.

9. Originally Posted by GustavoB
What sounds strange to me is that he adds a (r + 1) to both sides of the equation, then it became r(r + 1) + 2(r+1) and so on.

I don't understand this process.

Prove it already tried to explain me but i still can't see why to do a Sum S.

This page is a part of the book where the author tries to explain Proof by induction.

i think it must have been this way in the book
define:
f(r) = 1+2+3+...+r
to prove that f(r)= r*(r+1)/2

induction:
f(1) is true
assume f(r) is true.
then f(r+1) = f(r) +(r+1). do you see why?
so f(r+1)= r(r+1)/2 + (r+1)
so f(r+1)= (r+1)[r/2 + 1]
so f(r+1)= (r+1)(r+2)/2
so f(r+1) is also true.