Fibonacci Beweis f(n+1)^2 -fn^2=(f(n+2))(f(n-1))

Question

Fibonacci. Beweise:

f_n+1² -f_n²=(f_n+2)(f_n-1)

Answer 1

Nach Definition gilt \(f_{n+1}=f_n+f_{n-1}\).$$\Leftrightarrow f_{n+1}-f_n=f_{n-1}$$$$\Leftrightarrow (f_{n+1}-f_n)\cdot f_{n+2}=f_{n-1}\cdot f_{n+2}$$$$\Leftrightarrow (f_{n+1}-f_n)\cdot(f_{n+1}+f_n)=f_{n-1}\cdot f_{n+2}$$$$\Leftrightarrow f_{n+1}^2-f_n^2=f_{n-1}\cdot f_{n+2}.$$