The Lucas numbers $L_n$ are defined by the equations $L_1 = 1$, and $L_n = F_{n+1} + F_{n-1}$ for each $n \geq 2$.
What is wrong with the following argument?
Assuming $L_n = F_n$ for $n = 1,2,\cdots,k$, we see $L_{k+1} = L_k + L_{k-1}$ from an induction proof of the definition, $L_k + L_{k-1} = F_k + F_{k-1}$ by assumption, and $F_k + F_{k-1} = F_{k+1}$ by definition. This means $F_{k+1} = L_{k+1}$ so it's true for all positive $n$.
My attempt: The argument is not incorrect for the assumption. However, if we substitute the definition, we see $L_k + L_{k-1} = F_{k+1} + F_{k-1} + F_{k} + F_{k-2} \neq F_k + F_{k-1}$, so the argument is false.
You’ve shown that the conclusion of the argument is false; the argument must therefore be incorrect in some way, and you’re supposed to figure out where it goes astray.
In order for the induction step to be valid, we need to know that $L_k=F_k$ and $L_{k-1}=F_{k-1}$. That means that $k$ must be at least $2$, since $L_0$ has not been defined. Is it true that $L_2=F_2$ and $L_1=F_1$?