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Fibonacci Workings

0

$begingroup$

The Fibonacci number sequence is given by

$$F_n = F_n−1 + F_n−2, n ge 2,\F_0 = F_1 = 1$$

Suppose we write two consecutive Fibonacci numbers in the form of a $2 times 1$ matrix $f$, where

$$
f =
beginpmatrix
f_n\
f_n + 1\
endpmatrix.
$$

Show that there is a $2 times 2$ matrix $A$ such that when $f$ is multiplied by $A$ we get

$$
Abeginpmatrix
f_n\
f_n + 1\
endpmatrix
= beginpmatrix
f_n\
f_n + 1\
endpmatrix.
$$

Hint: What is $F_n+1$ in terms of $F_n$ and $F_n−1$?

induction fibonacci-numbers data-structure

share|cite|improve this question

edited Apr 2 at 18:48

Aaron Long

asked Apr 2 at 17:54

Aaron LongAaron Long

33

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  See en.wikipedia.org/wiki/Fibonacci_number#Matrix_form
  $endgroup$
  – lhf
  Apr 2 at 18:02
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  and this
  $endgroup$
  – J. W. Tanner
  Apr 2 at 18:07
  

  
  
  
  


• 
  
  
  

  
  6
  

  
  
  
  
  
  

  
  $begingroup$
  As written, A is the identity.
  $endgroup$
  – Doug M
  Apr 2 at 18:23
  
  

  
  
  
  

add a comment | 

0

$begingroup$

The Fibonacci number sequence is given by

$$F_n = F_n−1 + F_n−2, n ge 2,\F_0 = F_1 = 1$$

Suppose we write two consecutive Fibonacci numbers in the form of a $2 times 1$ matrix $f$, where

$$
f =
beginpmatrix
f_n\
f_n + 1\
endpmatrix.
$$

Show that there is a $2 times 2$ matrix $A$ such that when $f$ is multiplied by $A$ we get

$$
Abeginpmatrix
f_n\
f_n + 1\
endpmatrix
= beginpmatrix
f_n\
f_n + 1\
endpmatrix.
$$

Hint: What is $F_n+1$ in terms of $F_n$ and $F_n−1$?

induction fibonacci-numbers data-structure

share|cite|improve this question

edited Apr 2 at 18:48

Aaron Long

asked Apr 2 at 17:54

Aaron LongAaron Long

33

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  See en.wikipedia.org/wiki/Fibonacci_number#Matrix_form
  $endgroup$
  – lhf
  Apr 2 at 18:02
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  and this
  $endgroup$
  – J. W. Tanner
  Apr 2 at 18:07
  

  
  
  
  


• 
  
  
  

  
  6
  

  
  
  
  
  
  

  
  $begingroup$
  As written, A is the identity.
  $endgroup$
  – Doug M
  Apr 2 at 18:23
  
  

  
  
  
  

add a comment | 

$begingroup$

The Fibonacci number sequence is given by

$$F_n = F_n−1 + F_n−2, n ge 2,\F_0 = F_1 = 1$$

Suppose we write two consecutive Fibonacci numbers in the form of a $2 times 1$ matrix $f$, where

$$
f =
beginpmatrix
f_n\
f_n + 1\
endpmatrix.
$$

Show that there is a $2 times 2$ matrix $A$ such that when $f$ is multiplied by $A$ we get

$$
Abeginpmatrix
f_n\
f_n + 1\
endpmatrix
= beginpmatrix
f_n\
f_n + 1\
endpmatrix.
$$

Hint: What is $F_n+1$ in terms of $F_n$ and $F_n−1$?

induction fibonacci-numbers data-structure

share|cite|improve this question

edited Apr 2 at 18:48

Aaron Long

asked Apr 2 at 17:54

Aaron LongAaron Long

33

$endgroup$

share|cite|improve this question

edited Apr 2 at 18:48

Aaron Long

asked Apr 2 at 17:54

Aaron LongAaron Long

33

share|cite|improve this question

edited Apr 2 at 18:48

Aaron Long

asked Apr 2 at 17:54

Aaron LongAaron Long

33

asked Apr 2 at 17:54

Aaron LongAaron Long

33

asked Apr 2 at 17:54

Aaron LongAaron Long

33