Recurrence Relation


by Char. Limit
Tags: recurrence, relation
Char. Limit
Char. Limit is offline
#1
Sep27-10, 04:28 AM
PF Gold
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P: 1,930
1. The problem statement, all variables and given/known data
Let's say I had this recurrence relation:

[tex]log\left(f\left(x+2\right)\right) = log\left(f\left(x+1\right)\right) + log\left(f\left(x\right)\right)[/tex]

How do I prove, then, that...

[tex]f\left(x\right) = e^{c_1 L_x + c_2 F_x}[/tex]

?

2. Relevant equations

There probably are some, but I don't know any.

3. The attempt at a solution

I've gotten the equation to remove the logs, but I just get...

[tex]f\left(x+2\right) = f\left(x+1\right)f\left(x\right)[/tex]

I don't know where to go from there.
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HallsofIvy
HallsofIvy is offline
#2
Sep27-10, 06:54 AM
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Thanks
PF Gold
P: 38,882
First, use the properties of the logarithm to get rid of the logarithm:
[tex]log(f(x+ 2))= log(f(x+1))+ log(f(x))= log(f(x+1)f(x))[/tex]
and, since log is one-to-one, f(x+2)= f(x+1)f(x).

It's certainly true that the formula you gives satisfies that. Can you prove the solution is unique?


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