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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.

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.

Here are more sites:

http://www.mathsisfun.com/algebra/mathe … ction.html

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In mathematics, you don't understand things. You just get used to them.
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