+0  
 
0
340
3
avatar

What is the relationship of nϕ and ϕ^n, when n=1, 2, 3, 4, ..., 15

Guest Aug 22, 2017
 #1
avatar+20011 
0

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:

\(\small{ \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|} \hline F_{-2} & F_{-1} & F_0 & F_1 & F_2 & F_3 & F_4 & F_5 & F_6 & F_7 & F_8 &F_9 &F_{10} &F_{11} &F_{12} &F_{13} &F_{14} &F_{15} \\ \hline -1 & 1 & 0 & 1 & 1 & 2 & 3 & 5 & 8 & 13 & 21 & 34 & 55 & 89 & 144 & 233 & 377 & 610 \\ \hline \end{array} }\)

 

\(\begin{array}{|rcll|} \hline \text{relationship } &=& \frac{n\phi}{\phi^n} \\ &=& \frac{n\phi}{F_{n}\phi +F_{n-1}} \\ &=& \frac{n}{ \frac{F_{n}\phi +F_{n-1} } {\phi} } \\ &=& \frac{n}{ F_{n} + \frac{1}{\phi} F_{n-1} } \\ &=& \frac{n}{ F_{n} + (\phi -1) F_{n-1} } \\ &=& \frac{n}{ F_{n} + F_{n-1}\phi -F_{n-1} } \\ &=& \frac{n}{ F_{n}-F_{n-1} + F_{n-1}\phi } \\ &\mathbf{=}& \mathbf{\frac{n}{ F_{n-2} + F_{n-1}\phi }} \\ \hline \end{array}\)

 

\(\begin{array}{|r|l|} \hline \mathbf{n} & \mathbf{\frac{n\phi}{\phi^n} = \frac{n}{ F_{n-2} + F_{n-1}\phi }} \\ \hline 1 & \frac{ 1 }{ 1+0\phi } = 1 \\ 2 & \frac{ 2 }{ 0+1\phi } = \frac{2}{\phi}= 2(\phi -1) \\ 3 & \frac{ 3 }{ 1+1\phi } \\ 4 & \frac{ 4 }{ 1+2\phi } \\ 5 & \frac{ 5 }{ 2+3\phi } \\ 6 & \frac{ 6 }{ 3+5\phi } \\ 7 & \frac{ 7 }{ 5+8\phi } \\ 8 & \frac{ 8 }{ 8+13\phi } \\ 9 & \frac{ 9 }{ 13+21\phi } \\ 10 & \frac{10 }{ 21+34\phi } \\ 11 & \frac{11 }{ 34+55\phi } \\ 12 & \frac{12 }{ 55+89\phi } \\ 13 & \frac{13 }{ 89+144\phi } \\ 14 & \frac{14 }{ 144+233\phi } \\ 15 & \frac{15 }{ 233+377\phi } \\ \hline \end{array}\)

 

 

laugh

heureka  Aug 22, 2017
 #2
avatar
0

Heureka, how did you get that ϕ^n=Fnϕ+Fn-1?

Guest Aug 23, 2017
 #3
avatar+20011 
0

Heureka, how did you get that ϕ^n=Fnϕ+Fn-1 ?

 

Formula:  \( \phi^2 = \phi +1\)

mathematical proof:

\(\begin{array}{|rcll|} \hline \phi = \frac{1+\sqrt{5}} {2} \\\\ \phi^2 &=& \left( \frac{1+\sqrt{5}} {2} \right)^2 \\ &=& \frac{\left( 1+\sqrt{5}\right)^2 } {4} \\ &=& \frac{ 1+2\sqrt{5}+5 } {4} \\ &=& \frac{ 6+2\sqrt{5} } {4} \\ &=& \frac{ 3+\sqrt{5} } {2} \\ &=& \frac{ 1+2+\sqrt{5} } {2} \\ &=& \frac{ 1+\sqrt{5}+2 } {2} \\ &=& \frac{ 1+\sqrt{5} } {2} +\frac{2}{2} \\ &=& \frac{ 1+\sqrt{5} } {2} + 1 \\ &=& \mathbf{ \phi + 1 } \\ \hline \end{array}\)

 

\(\small{ \begin{array}{|lclclclcl|} \hline \phi^1&&&& &=& 1\phi+0 &=& F_1\phi + F_0 \\ \phi^2&&&& &=& 1\phi+1 &=& F_2\phi + F_1 \\ \phi^3 = \phi\phi^2 = \phi(\phi+1) &=&1\phi^2 + 1\phi &=&1(\phi+1)+1\phi &=& 2\phi+1 &=& F_3\phi + F_2 \\ \phi^4 = \phi\phi^3 = \phi(2\phi+1) &=&2\phi^2 + 1\phi &=&2(\phi+1)+1\phi &=& 3\phi+2 &=& F_4\phi + F_3 \\ \phi^5 = \phi\phi^4 = \phi(3\phi+2) &=&3\phi^2 + 2\phi &=&3(\phi+1)+2\phi &=& 5\phi+3 &=& F_5\phi + F_4 \\ \phi^6 = \phi\phi^5 = \phi(5\phi+3) &=&5\phi^2 + 3\phi &=&5(\phi+1)+3\phi &=& 8\phi+5 &=& F_6\phi + F_5 \\ \phi^7 = \phi\phi^6 = \phi(8\phi+5) &=&8\phi^2 + 5\phi &=&8(\phi+1)+5\phi &=& 13\phi+8 &=& F_7\phi + F_6 \\ \phi^8 = \phi\phi^7 = \phi(13\phi+8) &=&13\phi^2 + 8\phi &=&13(\phi+1)+8\phi &=& 21\phi+13 &=& F_8\phi + F_7 \\ \phi^9 = \phi\phi^8 = \phi(21\phi+13) &=&21\phi^2 + 13\phi &=&21(\phi+1)+13\phi &=& 34\phi+21 &=& F_9\phi + F_8 \\ \phi^{10} = \phi\phi^9 = \phi(34\phi+21) &=&34\phi^2 + 21\phi &=&34(\phi+1)+21\phi &=& 55\phi+34 &=& F_{10}\phi + F_9 \\ \cdots \\ \mathbf{ \phi^n } &&&& && &\mathbf{ =}& \mathbf{ F_n\phi + F_{n-1}} \\ \hline \end{array} }\)

 

 

laugh

heureka  Aug 23, 2017

17 Online Users

avatar

New Privacy Policy

We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive information about your use of our website.
For more information: our cookie policy and privacy policy.