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:

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- April 16th 2011, 10:27 PM #1

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- April 17th 2011, 12:52 AM #2

- April 17th 2011, 01:38 AM #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.

- April 17th 2011, 03:03 AM #4

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- April 17th 2011, 03:46 AM #5

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- April 17th 2011, 03:54 AM #6

- April 17th 2011, 04:31 AM #7

- April 17th 2011, 06:34 AM #8

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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.

- April 17th 2011, 06:40 AM #9

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.