Let $S_n=1+q+q^2+\ldots+q^n$ and $T_n=1+\left(\frac{q+1}{2}\right)+\left(\frac{q+1}{2}\right)^2+\ldots .+\left(\frac{q+1}{2}\right)^n$ where $q$ is a real number and $q \neq 1$. If ${ }^{101} \mathrm{C}_1+{ }^{101} \mathrm{C}_2 \cdot \mathrm{~S}_1+\ldots+{ }^{101} \mathrm{C}_{101} \cdot \mathrm{~S}_{100}=\alpha \mathrm{T}_{100}$.