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#1 2012-11-06 18:07:42

hempy
Member
Registered: 2012-11-06
Posts: 2

Induction Proofs

Let a(sub0) and r be fixed real numbers with r ≠ 0 and r ≠ 1, and suppose that for each n ∈ N, a(subn) = r*a(subn-1).

For every nonnegative integer n, a(subn) = a(sub0) * r^n.

Prove by induction.

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#2 2012-11-06 19:53:11

bob bundy
Moderator
Registered: 2010-06-20
Posts: 6,269

Re: Induction Proofs

hi hempy,

The following site has an explanation of proof by induction.  The final example is the proof for the geometric sequence.  The notation is a little different but you should be able to 'convert it'.

http://www.csee.umbc.edu/~stephens/203/PDF/4-2.pdf

Bob


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#3 2012-11-07 19:22:40

noelevans
Member
Registered: 2012-07-20
Posts: 236

Re: Induction Proofs

Hi!

Suppose that a   is a constant and r is not zero or one and for each n in {1,2,3,...}
                     0
                                                    n
a   = r*a     .    Show that a  = a * r    for each n in {1,2,3,...}.
  n         n-1                      n     0
                                                             1
For n=1 we have a  = r*a     = r*a  = a *1  .   So it works for n=1.
                           1        1-1        0     0

Now suppose that for some positive integer k,

                          k                                                               k               k          k+1
we have a  = a *r  .   Then  a      = r*a           = r*a   = r*a *r  = a * r * r   = a*r
               k     0                   k+1        (k+1)-1        k         0         0                0

So given that it works for k we have shown that it works for k+1, which is the inductions step.
Therefore it works for all n in {1,2,3,...}.

Is there any way to put this page into a half-space mode?  Then the super and subscripts would
look pretty nice.  smile


Writing "pretty" math (two dimensional) is easier to read and grasp than LaTex (one dimensional).
LaTex is like painting on many strips of paper and then stacking them to see what picture they make.

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#4 2012-11-07 20:21:42

bob bundy
Moderator
Registered: 2010-06-20
Posts: 6,269

Re: Induction Proofs

hi noelevans

You could try Latex:

http://www.mathisfunforum.com/viewtopic.php?id=4397

smile

Bob


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#5 2012-11-08 01:33:33

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 14,885

Re: Induction Proofs

The forum also has built in tags which allows use of subscripts and superscripts withouth knowledge of latex:

For example a[sub]n[/sub].


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#6 2012-11-08 04:47:16

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 83,138

Re: Induction Proofs

Hi noelevans;

Give this a try:

http://latex.codecogs.com/editor.php

perfect latex everytime!


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

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#7 2012-11-08 07:25:25

noelevans
Member
Registered: 2012-07-20
Posts: 236

Re: Induction Proofs

Thanks all! smile


Writing "pretty" math (two dimensional) is easier to read and grasp than LaTex (one dimensional).
LaTex is like painting on many strips of paper and then stacking them to see what picture they make.

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#8 2012-11-08 08:25:50

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 14,885

Re: Induction Proofs

Pretty math?


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#9 2012-11-08 13:29:58

noelevans
Member
Registered: 2012-07-20
Posts: 236

Re: Induction Proofs

LaTex input (not so pretty) vs LaTex output = Pretty math


Writing "pretty" math (two dimensional) is easier to read and grasp than LaTex (one dimensional).
LaTex is like painting on many strips of paper and then stacking them to see what picture they make.

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