Question

I. Determine whether each of the following statements about Fibonacci numbers is true or false. a. 2F₁>Fn+1for n ≥ 3 b. 2F +4 = Fn+3 for n ≥ 3

Answer 1

The Fibonacci sequence, $F_0, F_1,F_2, … ,$ is defined by the initial condition $F_0=0,$ $F_1=1,$ and the recurrence relation $F_n = F_{n-1}+F_{n-2}$ for $n=2, 3, 4, ...$

$$0,1,1,2,3,5,8,13,21,34,55,89,144,...$$

A.

$$F_{n+1}=F_{n}+F_{n-1}<F_n+F_n=2F_n ,n\geq3$$

The statement $2F_n>F_{n+1}$ is true for $n\geq 3$

B.

$$2F_{n+4}=2(F_{n+3}+F_{n+2})=F_{n+3}+F_{n+3}+2F_{n+2}$$

Since $F_k>0$ for $k\geq1,$ then

$$2F_{n+4}=F_{n+3}+F_{n+3}+2F_{n+2}>F_{n+3}, n\geq3$$

The statement $2F_{n+4}=F_{n+3}$ is false for $n\geq 3.$