Let a<b be real numbers ,define $\displaystyle x_n$ as follows:
$\displaystyle x_1=a \ , \ x_2=b \ and \ x_n = \frac{2}{3} x_{n-1} + \frac{1}{3} x_{n-2}\ , n \geq 3 $
Show that $\displaystyle x_n$ is convergent , what is its limit ?
Prove by induction that:
1) In reduced fractions we have $\displaystyle \forall\,n\,,\,\,x_n=\frac{A_n}{3^{n-2}}\,a + \frac{B_n}{3^{n-2}}\,b$ , for some positive $\displaystyle A_n\,,\,B_n<3^{n-2}$
2) $\displaystyle \forall\,n>2\,,\,\,A_n=B_{n-1}\,,\,A_n+B_n=3^{n-2}\Longrightarrow\,\frac{A_n}{3^{n-2}}+\frac{B_n}{3^{n-2}}=1$
Tonio