# Fibonacci inequality

#### greg1313

For $$\displaystyle \ n \ge 3, \ \$$ prove that

$$\displaystyle F_n + F_{n + 2} \ge 2F_{n + 1} + 1$$

Last edited by a moderator:

#### Plato

MHF Helper
For $$\displaystyle \ n \ge 3, \ \$$ prove that
$$\displaystyle F_n \ + \ F_{n + 2} \ \ge \ 2F_{n + 1} \ + \ 1$$
Hint:
I hope that you know that $F_1=1,~F_2=1~\&~n\ge 3 \Rightarrow F_{n}=F_{n-1}+F_{n-2}$
Use induction: is this true for the base case, $n=3~?$
Suppose that it is true for $n=K>3$. then look at $F_{K+1}=F_{(K+1)-1}+F_{(K+1)-2}$

Last edited by a moderator:

#### topsquark

Forum Staff
Plato: This is the Challenge Forum. He knows how to solve it.

-Dan