+26367 Relationship between n*ϕ and ϕ^n
What is the relationship of nϕ and ϕ^n, when n=1, 2, 3, 4, ..., 15
Formula:
\(\begin{array}{rcll} \phi^n &=& F_{n}\phi +F_{n-1}\\ F_{n-2} &=& F_{n}-F_{n-1}\\ \frac{1}{\phi} &=& \phi -1 \\ \end{array}\)
List of Fibonacci numbers:
\(\small{ \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|} \hline F_{-2} & F_{-1} & F_0 & F_1 & F_2 & F_3 & F_4 & F_5 & F_6 & F_7 & F_8 &F_9 &F_{10} &F_{11} &F_{12} &F_{13} &F_{14} &F_{15} \\ \hline -1 & 1 & 0 & 1 & 1 & 2 & 3 & 5 & 8 & 13 & 21 & 34 & 55 & 89 & 144 & 233 & 377 & 610 \\ \hline \end{array} }\)
\(\begin{array}{|rcll|} \hline \text{relationship } &=& \frac{n\phi}{\phi^n} \\ &=& \frac{n\phi}{F_{n}\phi +F_{n-1}} \\ &=& \frac{n}{ \frac{F_{n}\phi +F_{n-1} } {\phi} } \\ &=& \frac{n}{ F_{n} + \frac{1}{\phi} F_{n-1} } \\ &=& \frac{n}{ F_{n} + (\phi -1) F_{n-1} } \\ &=& \frac{n}{ F_{n} + F_{n-1}\phi -F_{n-1} } \\ &=& \frac{n}{ F_{n}-F_{n-1} + F_{n-1}\phi } \\ &\mathbf{=}& \mathbf{\frac{n}{ F_{n-2} + F_{n-1}\phi }} \\ \hline \end{array}\)
\(\begin{array}{|r|l|} \hline \mathbf{n} & \mathbf{\frac{n\phi}{\phi^n} = \frac{n}{ F_{n-2} + F_{n-1}\phi }} \\ \hline 1 & \frac{ 1 }{ 1+0\phi } = 1 \\ 2 & \frac{ 2 }{ 0+1\phi } = \frac{2}{\phi}= 2(\phi -1) \\ 3 & \frac{ 3 }{ 1+1\phi } \\ 4 & \frac{ 4 }{ 1+2\phi } \\ 5 & \frac{ 5 }{ 2+3\phi } \\ 6 & \frac{ 6 }{ 3+5\phi } \\ 7 & \frac{ 7 }{ 5+8\phi } \\ 8 & \frac{ 8 }{ 8+13\phi } \\ 9 & \frac{ 9 }{ 13+21\phi } \\ 10 & \frac{10 }{ 21+34\phi } \\ 11 & \frac{11 }{ 34+55\phi } \\ 12 & \frac{12 }{ 55+89\phi } \\ 13 & \frac{13 }{ 89+144\phi } \\ 14 & \frac{14 }{ 144+233\phi } \\ 15 & \frac{15 }{ 233+377\phi } \\ \hline \end{array}\)
+26367 Heureka, how did you get that ϕ^n=Fnϕ+Fn-1 ?
Formula: \( \phi^2 = \phi +1\)
mathematical proof:
\(\begin{array}{|rcll|} \hline \phi = \frac{1+\sqrt{5}} {2} \\\\ \phi^2 &=& \left( \frac{1+\sqrt{5}} {2} \right)^2 \\ &=& \frac{\left( 1+\sqrt{5}\right)^2 } {4} \\ &=& \frac{ 1+2\sqrt{5}+5 } {4} \\ &=& \frac{ 6+2\sqrt{5} } {4} \\ &=& \frac{ 3+\sqrt{5} } {2} \\ &=& \frac{ 1+2+\sqrt{5} } {2} \\ &=& \frac{ 1+\sqrt{5}+2 } {2} \\ &=& \frac{ 1+\sqrt{5} } {2} +\frac{2}{2} \\ &=& \frac{ 1+\sqrt{5} } {2} + 1 \\ &=& \mathbf{ \phi + 1 } \\ \hline \end{array}\)
\(\small{ \begin{array}{|lclclclcl|} \hline \phi^1&&&& &=& 1\phi+0 &=& F_1\phi + F_0 \\ \phi^2&&&& &=& 1\phi+1 &=& F_2\phi + F_1 \\ \phi^3 = \phi\phi^2 = \phi(\phi+1) &=&1\phi^2 + 1\phi &=&1(\phi+1)+1\phi &=& 2\phi+1 &=& F_3\phi + F_2 \\ \phi^4 = \phi\phi^3 = \phi(2\phi+1) &=&2\phi^2 + 1\phi &=&2(\phi+1)+1\phi &=& 3\phi+2 &=& F_4\phi + F_3 \\ \phi^5 = \phi\phi^4 = \phi(3\phi+2) &=&3\phi^2 + 2\phi &=&3(\phi+1)+2\phi &=& 5\phi+3 &=& F_5\phi + F_4 \\ \phi^6 = \phi\phi^5 = \phi(5\phi+3) &=&5\phi^2 + 3\phi &=&5(\phi+1)+3\phi &=& 8\phi+5 &=& F_6\phi + F_5 \\ \phi^7 = \phi\phi^6 = \phi(8\phi+5) &=&8\phi^2 + 5\phi &=&8(\phi+1)+5\phi &=& 13\phi+8 &=& F_7\phi + F_6 \\ \phi^8 = \phi\phi^7 = \phi(13\phi+8) &=&13\phi^2 + 8\phi &=&13(\phi+1)+8\phi &=& 21\phi+13 &=& F_8\phi + F_7 \\ \phi^9 = \phi\phi^8 = \phi(21\phi+13) &=&21\phi^2 + 13\phi &=&21(\phi+1)+13\phi &=& 34\phi+21 &=& F_9\phi + F_8 \\ \phi^{10} = \phi\phi^9 = \phi(34\phi+21) &=&34\phi^2 + 21\phi &=&34(\phi+1)+21\phi &=& 55\phi+34 &=& F_{10}\phi + F_9 \\ \cdots \\ \mathbf{ \phi^n } &&&& && &\mathbf{ =}& \mathbf{ F_n\phi + F_{n-1}} \\ \hline \end{array} }\)
