Recurrence Relation

  1. Char. Limit

    Char. Limit 1,986
    Gold Member

    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.
     
  2. jcsd
  3. HallsofIvy

    HallsofIvy 40,664
    Staff Emeritus
    Science Advisor

    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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