Answer
See explanation
Work Step by Step
## Identity to Prove
For all integers \(k \ge 1\),
\[
F_{k+1}^2 \;-\; F_k^2
\;=\;
F_{k-1}\,F_{k+2},
\]
where \(\{F_n\}\) is the Fibonacci sequence, defined by
\[
F_0 = 0,\quad F_1 = 1,\quad \text{and} \quad F_{n} = F_{n-1} + F_{n-2}\ \text{for}\ n \ge 2.
\]
---
## Proof
1. **Rewrite the left‐hand side by factorization:**
\[
F_{k+1}^2 - F_k^2
\;=\;
(F_{k+1} - F_k)\,\bigl(F_{k+1} + F_k\bigr).
\]
2. **Substitute known Fibonacci relations:**
- From \(F_{k+1} = F_k + F_{k-1}\), it follows that \(F_{k+1} - F_k = F_{k-1}\).
- From \(F_{k+2} = F_{k+1} + F_k\), it follows that \(F_{k+1} + F_k = F_{k+2}\).
Hence,
\[
(F_{k+1} - F_k)\,\bigl(F_{k+1} + F_k\bigr)
\;=\;
F_{k-1} \,\cdot F_{k+2}.
\]
Putting these together gives
\[
F_{k+1}^2 - F_k^2 = F_{k-1}\,F_{k+2},
\]
as required.
