Mathematical induction - how-to
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Mathematical induction - how-to

[From: ] [author: ] [Date: 11-05-08] [Hit: ]
QED-n is either odd or n is even.If n is even then n^2 is also evenAn even number plus an even number equals an even number, so is divisable by 2If n is odd, then n^2 is also oddAn odd number plus an odd number equals an even number,......
Try for n = 2 :
2^2 + 2 = 4 +2 = 6 again... yes it does ..
etc, etc ..

So: the induction proof :
lets assume it works for some n : Then (n^2 +n) /2 evenly !
now try it for n ---> n+1
( by subbing in (n+1 ) for the n )

then for n---> n+1 we get :
(n+1)^2 + (n+1 ) = n^2 +2n +1 + n +1
= n^2 +3n + 2

Now we break THIS apart, by splitting up the 3n into two parts :

n^2 +3n + 2 = [ n^2 +n] + [2n +2 ]

Taking it term by term:
[ n^2 + n ] does divide by 2 ( original assumption of the induction proof)
and for the other term, [ 2n+2 ] this factors into 2[n+1 ] which also divides by 2 ( at least the '2' part does. )

so both terms divide by 2
thus if it works for n, then it works for n+1...
as in :
it works for 2 ( Vide supra ) so it works for 3
It works for 3 so it works for 4
It works for 4 so it works for 5 ..... etc , etc, etc up to infinity ..

QED

-
n is either odd or n is even.

If n is even then n^2 is also even

An even number plus an even number equals an even number, so is divisable by 2

If n is odd, then n^2 is also odd

An odd number plus an odd number equals an even number, so is divisable by 2
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