The Fibonacci sequence $F_0, F_1, F_2,...,$ is defined by the rule:$$F_0=0, F_1=1, F_n=F_{n-1}+F_{n-2}$$ Use induction to prove that $F_n\geq2^{0.5n}$ for $n\geq 6$
So far I have done the basis step plugging in $6$ and getting $8$ in return.Next I do the inductive step now I have $F_{k+1}=F_k+F_{k-1}$ and use the $F_n\geq2^{0.5n}$ they gave me I end up with $$2^{\frac{n}{2}}+2^{\frac{n-1}{2}}$$at which point I get stuck I try and simplify the expression but what I end up with is different from the solutions. Can someone walk me throught this proof and explain how to do it please?
