Math Is Fun Forum

frankitox

Member

Registered: 2011-12-04

Posts: 4

Quick Tip on Proof Sum of Squares

First of all, I was looking for proofs for this:

I check it by induction, but I see on other topic of the forum this one:

I didn't get it from the 3rd to 4th line.
In the right side use distribution, but on the left
Can anybody 'expand it' for me?

Ps: It's my first post so if there is something wrong, just tell me. And sorry for my bad english.

Edit: Here is the old post http://www.mathisfunforum.com/viewtopic.php?id=2014

Last edited by frankitox (2011-12-04 01:38:57)

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Quick Tip on Proof Sum of Squares

Hi;

That step is correct. I have bracketed it off to make it clearer.

Welcome to the forum.

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,140

Re: Quick Tip on Proof Sum of Squares

hi frankitox

Welcome to the forum.

I'll go through it line by line.

I guess you're ok with the standard binomial expansion.

Then switch about the terms.

Now the next line should have had some brackets as everything is being 'summed'.  And some limits!

Now there a line or two missing which probably explains the confusion!

And a sneaky change of variable and limits.  In the second term replace x by x+1 and change the limits to compensate.

Now that allows us to do some cancelling. 

Every term except the last from the first expression and the first from the second expression cancels out leaving

Notice that 'n' on the left hand side; not an 'x' any longer.

Now we can sort out the right hand side; by expanding into separate sums:

We have a formula already for sum of 'n' numbers and adding up '1' n-times just makes n.

Oh yes, and let's factorise the 3 while we're at it.

So rearrange:

and so finally:

Hope that helps. 

Bob

ps.  Induction is good for these formulas but you have to know what the formula is before you can start.
This method gives the formula from scratch.
And a similar method will work for the sum of cubes (start with (x+1)^4) and the sum of power fours (start with (x+1)^5) ...etc.

pps.  Bad English?  Where was that then?

Last edited by Bob (2011-12-04 05:32:22)

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

frankitox

Member

Registered: 2011-12-04

Posts: 4

Re: Quick Tip on Proof Sum of Squares

Hi Bob! I'm feedbacking you really late! Sorry for that. First, I have to say thanks for the answer and your time.
Yes, I love this method for the reason that you said!
Now.. I can't see the transition here..

IF you like to explain it to me. I don't know if i'm missing a property or something like that. You write down that you was using cancelling.
Really pissed me off that I can't see it!
Mr, you have a degree on maths, or studying, or hobby?
I really appreciate you're help. Thanks

frankitox

Member

Registered: 2011-12-04

Posts: 4

Re: Quick Tip on Proof Sum of Squares

I get it doing this

Finally! This is the way you make the cancelling? all that in your head??

Well, now i put here this resource
http://www.proofwiki.org/wiki/Sum_of_Sequence_of_Squares
(the graph is really cool)

Last edited by frankitox (2011-12-15 12:41:21)