question_answer

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 \[^{101}{{C}_{1}}{{+}^{101}}{{C}_{2}}.{{S}_{1}}+.....{{+}^{101}}{{C}_{101}}.{{S}_{100}}={{\alpha }^{100}}{{T}_{100}},\]then \[\alpha \] is equal to

Answer:

2