Prove whether or not any Fibonacci number, F(n), can be constructed by using
F(n) F(n-1)'s and the operators +,-,*,/
(Provided that F(n-1)>0 )
For example I can make a 5 using 5 3's: 3 + 3/3 + 3/3

For brevity denote F(n) by F  and  F(n-1) by P

Ans:

For ODD (the majority of Fibo series) values of F:
F=P+P/P added (F-P) times +(P-P )added (p-1/2*(F+1)) times

e.g. 13 =13+8/8 added (5) times+...(8-8)added (8-14/2) times
#OF TERMS=1+5*2+2*1=13

For EVEN(the rest of Fibo series) values of F:
F=P+P-P/P repeated (2*P-F) times +(P-P )added 1/2*(3*F-4*P-2) times

e.g. 34=21+21-21/21 repeated (8) times+...(21-21)added 1/2*(102-84-2) times
#OF TERMS=2+8*2+2*8=34

Works for any member except 1.

 

Edited on July 10, 2010, 6:09 am

Posted by Ady TZIDON on 2010-07-10 06:06:23