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#1 2016-09-19 06:02:32

markosheehan
Member
Registered: 2016-06-15
Posts: 51

induction

prove by induction for all positive integers n: 1+5+9+13+........+(4n-3)= n/2(4n-2)
i tried this by trying to prove n/2(4n-2)+ (4(k+1)-3) = k+1/2(4(k+1)-2) but it did not work out for me.

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#2 2016-09-19 08:39:06

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: induction

Hi;

Ain't done one of these in nearly a century. Back then I had quill pens and candles. Sometimes, I would grab the pigtails of the cute girl (Rosemary M) sitting in front of me and dip them in the inkwell. This would make her cry and the teacher, ( I had Dominican O.P. nuns for teachers ) very angry and she would beat me with a yardstick. Then I would have to pray, of course God knew I was not sorry. Just as he must have known, the next day I would do it again. Where is this going?  Back to induction...

Prove the first case when n = 1:

Now all we need to do is show that the nth case trips the (n+1)th.

We conjecture that

If that is true then the (n+1)th case ought to be true also.

Now according to 2)

the boxed off part equals

So 3) becomes:

Simplifying both sides:

So what happened here? If 2) the nth case is true then 3) the (n+1)th case is also true. Since we know that the first case when n = 1 is true by direct calculation we now know that n = 2 is true. Since n = 2 is true we know n = 3 is true...

We have proved 2) by induction, we are done.

domino-effect.jpg

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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.

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