Math Is Fun Forum

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.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,646

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

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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

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. 

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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.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,646

Re: Induction Proofs

hi noelevans

You could try Latex:

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

Bob

---

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

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

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].

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“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
The knowledge of some things as a function of age is a delta function.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Induction Proofs

Hi noelevans;

Give this a try:

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

perfect latex everytime!

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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.

noelevans

Member

Registered: 2012-07-20

Posts: 236

Re: Induction Proofs

Thanks all!

---

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.

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: Induction Proofs

Pretty math?

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“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
The knowledge of some things as a function of age is a delta function.

noelevans

Member

Registered: 2012-07-20

Posts: 236

Re: Induction Proofs

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

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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.