Is the sequence x(n) defined recursively as follows:
\[x(1)=1, x(2)=2;\] \[x(n)=\frac{x(n-1)+x(n-2)}{2}\]
monotone increasing?
monotone decreasing?
bounded above?
bounded below?
A sequence x(n) or real numbers is monotone increasing if \(x(n+1)\geq x(n)\) for all \(n\geq 1\).
At first we stated \(x(2) = 2 > x(1) =1\) so the sequence starts monotone increasing.
When x = 3 we get \[x(3) = \frac{x(3-1) +x(3-2)}{2}\] \[= \frac{x(2)+x(1)}{2}\] \[=\frac{2+1}{2}\] =\( \frac{3}{2}\)
Notice that \(x(2) > x(3) > x(1)\).
So x(n) is monotone increasing and decreasing for all \(n\geq 1\).