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Topic: Terms of a Sequence
Replies: 2   Last Post: Feb 21, 2007 11:50 AM

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R. Brown

Posts: 54
From: texas
Registered: 12/29/05
Terms of a Sequence
Posted: Feb 13, 2007 11:52 AM
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Ok, perhaps I should have extended the decimal numbers, including 1.4142135 & 2.828427) toward [infinity].... then what I stated would have been more correct. If everyone is going to quibble over such small matters then you are going to miss the real gist of all the Col's statements & questions & theories & conjectures, which are bound to shape mathematics of the future! I suggest you take a good, long & hard look at the following simple math statements/ideas:

B^{y}= (a + bi)_{y}....eq. 101

Log_{B}(a + bi)= y .....eq. 201

(a + bi)^{1/y}= B ......eq. 301

We see the immediate relationship between the 'rank' of a term from a geometrical sequence and the Logarithm of the base: y = y

Now, let's alter the equations to fit into a 3-axis graphing, where the 3 axis are: X,Z,(Y/i)

assume:
a= x
b= z

Then eq. 101 becomes:

B^{y}= (x + zi)....eq. 101-A

Then:
If (y) is the parameter, we can define (x) from the above eq. 101-A:

x= B^{Y} - zi

Then:
If (y) is the parameter, we can define (z) from the above eq. 101-A:

z= [B^{Y} - x]/i

The above simple statements fit nicely with my "principle of sums" concept and may eventually prove the following conjecture about parabolas/quadratics:

Rbtx's Conjecture-1001:
A parabola is a collection of ranked Terms in which no two terms share the same 'log base' in any geometric sequence. This is always true but with 1 possible exception only (per parabola). The exception being when a point has the coordinates: x=1,y=0, then 2 points may share the same Log Base.......Otherwise the following eq. is generally true for all quadratics of this form: AX^2 + BX + C = Y

X^{Y} =/= X'^{Y'}



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