Answer to Question #262439 in Discrete Mathematics for Ghk

Question

Determine whether this statement about Fibonacci numbers is true.

2𝐹𝒙 − 𝐹𝒙−2 = 𝐹𝒙+1 for x> 1 where 𝐹𝒙 is the 𝒙𝑡ℎ Fibonacci number.

Answer 1

Let $x$ any positive integer. If $F_x$ is what we use to describe the $x$th Fibonacci number, then

$$F_x = F_{x−1} + F_{x−2}$$

Then

$$F_{x+1} = F_{x} + F_{x−1}=F_{x}+(F_{x}-F_{x-2})$$

$$=2F_{x}-F_{x-2}$$

The statement $F_{x+1} =2F_{x}-F_{x-2}$ is true.