The electron configuration for potassium (\( \text{K} \), atomic number 19) is:
\(\text{K} : 1s^2, 2s^2, 2p^6, 3s^2, 3p^6, 4s^1.\)
The outermost orbital is the \( 4s \) orbital. The quantum numbers for the outermost electron are:
\( n = 4, \, l = 0, \, m = 0, \, s = +\frac{1}{2}. \)
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: