I have to prove $F^2_{n−1} = F^2_n + F^2_{n−1}$ for any $n >=1$ by induction (for the Fibonacci sequence).
For the basis step, I have:
$n = 1; $ $F_{(1)-1} = F^2_{(1)} + F^2_{(1)-1} ->$
$ 1 = 1^2+ 0$
As the base case holds, we can then assume that $n=k$:
$F_{2(k)-1} = F^2_{(k)} + F^2_{(k)-1} $
Next, this has to be proven for k + 1:
$F_{2(k+1)-1} = F^2_{(k+1)} + F^2_{(k+1)-1} $
I know that you have to somehow used the statement derived on the LHS of the equation in the previous step ( i.e. $F_{2(k)-1}$ ) when proving for $k+1$, but I'm not really sure how to apply the logic here since I've normally seen it used in cases where there is a summation on the LHS.
I currently have worked this out:
$F_{2(k+1) - 1} = F^2_{k + 1} + F^2_{(k+1) -1} ->$
$F_{2k + 1} = F^2_{(k+1)} + F^2_k$
and with some manipulation (I was thinking), you could factor out the $F^2$ from both terms on the RHS resulting in this:
$F^2_{(2k +1)}$, but even after doing this, the term on the LHS is:$F_{2k + 1}$ so even though both sides are similar, they differ by the squared term on the LHS.
Is my logic wrong in the inductive step? If so, how should I go about the proof?
