Let $r$ be a real number with $\vert r \vert < 1$ and let $$ S = \sum_{n = 0}^\infty r^n\,. $$ You may assume without proof that $S = \dfrac{1}{1 - r}\, $.

Let $p = 1 + r + r^2$. Sketch the graph of the function $1 + r + r^2$ and deduce that $\frac{3}{4} \le p < 3\,$.

Show that, if $1 < p < 3$, then the value of $ p$ determines $r$, and hence $S$, uniquely.

Show also that, if $\frac{3}{4} < p < 1$, then there are two possible values of $S$ and these values satisfy the equation $(3 - p)S^2 - 3S + 1 = 0$.

Let $r$ be a real number with $\vert r \vert < 1$ and let $$ T = \sum_{n = 1}^\infty nr^{n - 1} \,. $$ You may assume without proof that $ T = \dfrac{1}{(1 - r)^2}\,. $

Let $ q = 1 + 2r + 3r^2$. Find the set of values of $q$ that determine $T$ uniquely.

Find the set of values of $q$ for which $T$ has two possible values.

Find also a quadratic equation, with coefficients depending on $q$, that is satisfied by these two values.