Answer by R. Burton for Following Fibonacci
This is probably the least effective method possible, but there is a direct way to show this without induction. Let $f:\Bbb{R}\to\Bbb{C}$ where......
View ArticleAnswer by Pythagoras for Following Fibonacci
Using the notations and function $g$ given by J.G., define $$f(a,b)=a^4+2a^3b-a^2b^2-2ab^3+b^4.$$ It suffices to prove that $f(a,b)=1$. But $$f(a,b)=(a^2+ab-b^2)^2$$ and...
View ArticleAnswer by marty cohen for Following Fibonacci
To prove $F_ {n + 2} = \sqrt{\frac{F_n {F_ {n + 1} ^ 2}(3 {F_n} +4 {F_ {n + 1}}) + 1 }{{F_n} ^ 2 + {F_ {n + 1}} ^ 2}} $, I would write it as $F^2_{n + 2}(F_n ^ 2 + F_ {n + 1}^ 2) = F_n F_ {n + 1} ^ 2(3...
View ArticleAnswer by J.G. for Following Fibonacci
The hypothesis is equivalent to $g(n)=0$ with$$g(n):=F_nF_{n+1}^2(3F_n+4F_{n+1})+1-F_n^2F_{n+2}^2-F_{n+1}^2F_{n+2}^2\\=-F_n^4-2F_n^3F_{n+1}+F_n^2F_{n+1}^2+2F_nF_{n+1}^3-F_{n+1}^4+1.$$Since...
View ArticleFollowing Fibonacci
Prove that $ F_ {n + 2} = \sqrt{\frac{F_n {F_ {n + 1} ^ 2}(3 {F_n} +4 {F_ {n + 1}}) + 1 }{{F_n} ^ 2 + {F_ {n + 1}} ^ 2}} $? I discovered this property from an attempt to solve the following problem:...
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