Let r satisfy the equation r^2 = r + 1.
Show that the sequence s_n = ar^n, where A is constant, satisfies the fibonacci equation f_n = f_(n-1) + f_(n-2).
I'm not sure how to show this.
I know that f_1 = 1 & f_2 = 2
f_1 = 1
f_2 = 2
f_3 = f_2 + f_1
f_4 = f_3 + f_2
So
s_3 = s_2 + s_1 = Ar + Ar
s_4 = s_3 + s_2 = Ar + Ar + Ar
s_5 = s_4 + s_3 = Ar + Ar + Ar + Ar + Ar
s_6 = s_5 + s_4 = Ar + Ar + Ar + Ar + Ar + Ar + Ar + Ar
May be I can try this one ... It can be proved using induction
s_n = ar^n
so ...
s_0 = a
s_1 = ar
s_2 = ar^2 .... a+ar = a(1+r) = ar^2
similarly
s_k = ar^k .... ar^(k-1) + ar^(k-2) = (ar^(k-2)) * (1+r) = (ar^(k-2) ) * r^2 = ar^k
Cheers!
Sunnycoder