Question 125
Let $a_{1}, a_{2}, \ldots, a_{n}, \ldots$ be a geometric progression with $a_{1}=3-2 a$ and ratio $q=\frac{3-2 a}{a-2}$, where $a \neq \frac{3}{2}, 2$ is a real number. Set $S_{n}=\sum_{i=1}^{n} a_{i}$, $n \geq 1$. Prove that if the sequence $\left\{S_{n}\right\}_{n=1}^{\infty}$ is convergent and its limit is $S$, then $S\lt 1$.