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

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.

Induct on $n$, with the claim quantified over every $m\ge n$. At $n=0$,Suppose the identity holds for a fixed $n$ and every $m\ge n$. For $m\ge n+1$, the Fibonacci recurrence givesApplying the induction hypothesis with $m-1\ge n$ is legitimate. This proves $F_m{F}_{n+1}-F_n{F}_{m+1}=(-1{)}^n{F}_{m-n}$ throughout its domain, a case of the Fibonacci determinant identity.