The terms of the bibonacci sequence are given by
F0=0, F1=1 and F_n+2=F_n+1 +f_n for n> or equals to 0
thus each term is the sum of the previous two; the sequence begins 0,1,1,2,3,5,8.......
(a) for n=0,1,2,3, verify directly that f_n= 1/root 5 (a^n -b^n)
wher a =1/2(1+root 5 ) and b=1/2(1-root 5)
(b) by considernig the relations F_n+1= 1F_n+1 + 1F_n and
F_n+1= 1 F_n+1 +0F_n
prove that F_n=1/root 5 (((1+root5)^n/2) -(1-root 5)^n /2))
hint: you will probably find it easier i you keep the names a and b for the numbers 1/2(1 +root 5) and 1/2(1-root 5) until the end. note that these are the eigenvalues of the 2x2 matrix A(1 1
1 0 ) note also that 1x2 matrix (a with 1 under) and (b with 1 under) bre the eigenvectors of a. to confirm this u need to use the fact a^2=a+1 and b^2=b+1 ie. a and b are both roots of the charcteristic equation
so hard found in a bok need help please!
thanks !1