smc

Notice that $\frac{2F(n)+1}{2}=F(n)+\frac{1}{2}.$ This means that for every single increment $n$ goes up from $1$, $F(n)$ will increase by $\frac{1}{2}.$ Since $101$ is $100$ increments from $1$, $F(n)$ will increase $\frac{1}{2}\times 100=50.$ Since $F(1)=2,$ $F(101)$ will equal $2+50=\boxed{52}.$