Answer
See explanation
Work Step by Step
**Proof (direct expansion using the Fibonacci recurrence):**
We want to prove that for all integers \(k \ge 1\),
\[
F_{k+1}^2 \;-\; F_k^2 \;-\; F_{k-1}^2
\;=\;
2\,F_k\,F_{k-1}.
\]
Recall the Fibonacci recurrence: \(F_{k+1} = F_k + F_{k-1}\). Square both sides:
\[
F_{k+1}^2
\;=\;
(F_k + F_{k-1})^2
\;=\;
F_k^2 + 2\,F_k\,F_{k-1} + F_{k-1}^2.
\]
Now substitute this into the left‐hand side of the desired identity:
\[
F_{k+1}^2
\;-\;
F_k^2
\;-\;
F_{k-1}^2
\;=\;
\bigl(F_k^2 + 2\,F_k\,F_{k-1} + F_{k-1}^2\bigr)
\;-\;
F_k^2
\;-\;
F_{k-1}^2
\;=\;
2\,F_k\,F_{k-1}.
\]
Hence,
\[
F_{k+1}^2 - F_k^2 - F_{k-1}^2
\;=\;
2\,F_k\,F_{k-1},
\]
as was to be shown.
