Question

Random question #005

Answer 1

Let F_n be k^n.
k^n=2k^(n-1)+k^(n-2)
k^2=2k+1
k=1±sqrt(2)

F_n=A(1+sqrt(2))^n+B(1-sqrt(2))^n
F_1=A(1+sqrt(2))+B(1-sqrt(2))=1 ---(1)
F_2=A(3+2sqrt(2))+B(3-2sqrt(2))=1 ---(2)
(1): A(3+2sqrt(2))-B=1+sqrt(2) ---(3)
(2)-(3): B(4-2sqrt(2))=-sqrt(2)
B=-sqrt(2)/(4-2sqrt(2))=-(1+sqrt(2))/2
A=1/sqrt(2)-1/2

F_3=complicated calculation = 3
Confirmed.

Answer 2

\[F_n=\left(\frac1{\sqrt2}-\frac12\right)(1+\sqrt2)^n-\left(\frac{1+\sqrt2}2\right)(1-\sqrt2)^n\]

Answer 3

Now I have to go, so others please continue