Among the given species. \( \text{S}_2\text{O}_3^{2-} \) (thiosulfate ion) has a pyramidal geometry around the central sulfur atom due to the presence of a lone pair on sulfur, resulting in a trigonal pyramidal shape.\( \text{SO}_4^{2-} \) (sulfate ion) has a tetrahedral geometry due to its symmetrical distribution of oxygen atoms around the central sulfur atom.\( \text{SO}_3^{2-} \) (sulfite ion) also adopts a trigonal pyramidal structure around sulfur, but in this context, \( \text{S}_2\text{O}_3^{2-} \) is more relevant to pyramidal geometry.\( \text{S}_2\text{O}_7^{2-} \) (disulfate ion) has a tetrahedral arrangement with respect to its central atoms, making it non-pyramidal.
Therefore, only one species, \( \text{S}_2\text{O}_3^{2-} \), exhibits pyramidal geometry.
Let \( S = \left\{ m \in \mathbb{Z} : A^m + A^m = 3I - A^{-6} \right\} \), where
\[ A = \begin{bmatrix} 2 & -1 \\ 1 & 0 \end{bmatrix} \]Then \( n(S) \) is equal to ______.
Let \( T_r \) be the \( r^{\text{th}} \) term of an A.P. If for some \( m \), \( T_m = \dfrac{1}{25} \), \( T_{25} = \dfrac{1}{20} \), and \( \displaystyle\sum_{r=1}^{25} T_r = 13 \), then \( 5m \displaystyle\sum_{r=m}^{2m} T_r \) is equal to: