$$f(n) = \left\{\begin{matrix} 0 & n=1\\ 1 & n=2\\ f_{n-1} + f_{n-2} & n\geqslant 2\end{matrix}\right.$$
How can I prove by induction that $$f_{n} \geq \left ( 1.5 \right )^{n-1}$$ for all$$ n\geq l_{b}$$, I have to find the smallest value for $$l_{b}$$
Let $b = \min\{ n, f_n\ge 1.5^{n-1} \text{ and }f_{n+1}\ge 1.5^{n} \}$.
Let us prove that if $k\ge b$, $f_k\ge 1.5^{k-1}$ by induction. This is true for $b$ and $b+1$.
If this is true for $k$ and $k+1$ then $$ f_{k+2}\ge f_{k+1}+f_k \ge 1.5^{k} + 1.5^{k-1} = 1.5^{k-1}\times 2.5 \ge 1.5^{k-1}\times 2.25= 1.5^{k+1} $$
Then you find that $b=6$ using your calculator. Then $b=l_b$.
