Past exam of the year 1 of the computer science course of the University of Oxford / 2019 / Discrete mathematics, continuous mathematics, and probability / 2 / d / i / Solution

For $n=0$, the difference is $F_1^2-F_0{F}_2=1=(-1{)}^0$. Define $D_n=F_{n+1}^2-F_n{F}_{n+2}$. The Fibonacci recurrence givesThe inductive step therefore changes $(-1{)}^n$ into $(-1{)}^{n+1}$. By mathematical induction, $D_n=(-1{)}^n$ for all $n\ge 0$, the requested form of Cassini's identity.