Let $S_n=1+q+q^2+....+q^n$ and $T_n=1+\left(\frac{q+1}{2}\right)+{\left(\frac{q+1}{2}\right)}^2+......+{\left(\frac{q+1}{2}\right)}^n$ where q is a real number and $q\ne 1$.

If   ${}^{101}{C}_1{+}^{101}{C}_2.S_1+....{+}^{101}{C}_{101}.S_{100}=\alpha T_{100.}$, then $\alpha$ is equal to :-