Let $a ∈ R$ Consider $x_{1} = a, x_{2} = (1+a)/2$ , and by induction $x_{n} := (1+x_{n−1})/2$ What is the limit $?$
By replacing $x_{n}$ and $x_{n-1}$ by $l$, we get the limit $l=1$. So limit should be $1$.
Also the nth term can be represented by $x_{n} = ( a+ 1 + 2 + 2^{2}+......+ 2^{n-2})/ 2^{n-1}$ And again the limit is $1$
I want to know, why does the value of $a$ doesn't affect the limit$?$ Does this sequence always converge$?$ And also what can I say about the monotonicity of the sequence$?$