Register to reply

Recurrence Relation

by Char. Limit
Tags: recurrence, relation
Share this thread:
Char. Limit
#1
Sep27-10, 04:28 AM
PF Gold
Char. Limit's Avatar
P: 1,943
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.
Phys.Org News Partner Science news on Phys.org
Security CTO to detail Android Fake ID flaw at Black Hat
Huge waves measured for first time in Arctic Ocean
Mysterious molecules in space
HallsofIvy
#2
Sep27-10, 06:54 AM
Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 39,345
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?


Register to reply

Related Discussions
Recurrence Relation Calculus & Beyond Homework 1
Recurrence Relation Calculus & Beyond Homework 0
Recurrence Relation Help Calculus & Beyond Homework 5
Recurrence relation General Math 7
Recurrence Relation General Math 11