Fibonacci inequality

Dec 2016
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For \(\displaystyle \ n \ge 3, \ \ \) prove that

\(\displaystyle F_n + F_{n + 2} \ge 2F_{n + 1} + 1\)
 
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Plato

MHF Helper
Aug 2006
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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 $F_{K+1}=F_{(K+1)-1}+F_{(K+1)-2}$
 
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topsquark

Forum Staff
Jan 2006
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Plato: This is the Challenge Forum. He knows how to solve it.

-Dan