Let $0 < a < b < \infty$. Define $x_1=a$, $x_2=b$, and $x_{n+2} = \frac{x_n + x_{n+1}}{2}$ for $n \geq 1$. Does $\{x_n\}$ converge? If so, to what limit?
Does $x_{n+2} = (x_{n+1} + x_{n})/2$ converge?
1
$\begingroup$
real-analysis
-
0First find out what's going on. Then you can probably do it yourself. Put points $a$, $b$ on the $x$-axis. Where is $x_3$? Where is $x_4$? where is $x_5$? If at first you cant't do it for general $a$ and $b$, use $2$ and $15$. – 2012-11-28