Inequality for recursive-defined values

Question

$f_{n+2} = \frac{6}{5}f_{n+1}-f_{n}, f_0 = 0, f_1 = 1$ I need to prove that $f_n < 5/4$

I found that $f_{n} = \frac{1}{8} i 5^{1-n} \left((3-4 i)^n-(3+4 i)^n\right)$ and spend much time for efforts to solve this equality, but end in failure..

Answer 1

If we take $z=\frac{3-4i}{5}$ we have that: $$ f_n = \frac{5i}{8}(z^n-\bar{z}^n) $$ but since $z$ belongs to the unit circle so does $z^n$ and $|z^n-\bar{z}^n|$ cannot exceed $2$.