Question

x_(n+1)=[(x_n)/2]+1. here x_1=1. Does {x_n} converge? If yes, limit=?

Answer 1

basically the method involves making an educated guess as to what form the solution holds, and figuring out what coefficients make it work for the particular initial conditions. Now that you have a closed form for the sequence:\[x_n = -2\left(\frac{1}{2}\right)^n+2\]it chould be easy to see the sequence converges to 2 as n goes to infinity.

Answer 2

should*