Determine whether each following statements about Fibonacci numbers is true or false A. 2Fn >F n+1 for n≥3 B. 2F n+4 = f n+3 for n≥3

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 $n\geq 3.$