Originally Posted by

**pavon** Hi, I am trying to solve the following limit:

$\displaystyle \lim_{k\to\infty} \frac{1}{k} \log N(k)$

where

$\displaystyle N(k) = N(k-1) + N(k-2)$

with provided initial conditions. I have solved the recurrence relationship, and the specific solution does contain two terms:

$\displaystyle N(k) = a_1 r_1^k + a_2 r_2^k$

I've tried a few tricks to simplify the log expression, including:

$\displaystyle \frac{1}{k}logN(k) = \frac{1}{k}\log\left(a_1 r_1^k\left(1 + \frac{a_2}{a_1}\left(\frac{r_2}{r_1}\right)^k \right) \right) = \frac{1}{k}\log(a_1) + \log(r_1) + \log\left(1 + \frac{a_2}{a_1}\left(\frac{r_2}{r_1}\right)^k \right) $

and was able to modify this expression to bound the limit, but haven't been able to obtain the limit itself.

I have also tried to compute the limit directly on the recurrence relationship (rather than solving it first) but ran into similar problems with log(A+B) terms. Any suggestions on how to proceed?