Binomial Theorem and Its Simple Applications: JEE Main 2019 Mathematics question

Let S_n = 1 + q + q² + . . . . . + q^n and T_n = 1 + $\left( {{{q + 1} \over 2}} \right) + {\left( {{{q + 1} \over 2}} \right)^2}$ + . . . . . .+ ${\left( {{{q + 1} \over 2}} \right)^n}$ where q is a real number and q $\ne$ 1. If   ¹⁰¹C₁ + ¹⁰¹C₂ . S₁ + .... + ¹⁰¹C₁₀₁ . S₁₀₀ = $\alpha$T₁₀₀ then $\alpha$ is equal to