Four, $n^{2}(n \geqslant 4)$ positive numbers are arranged in $n$ rows and $n$ columns;
$$
\begin{array}{llllll}
a_{11} & a_{12} & a_{13} & a_{14} & \cdots & a_{1 n} \\
a_{21} & a_{22} & a_{23} & a_{24} & \cdots & a_{2 n} \\
a_{31} & a_{32} & a_{33} & a_{34} & \cdots & a_{3 n} \\
a_{41} & a_{42} & a_{43} & a_{44} & \cdots & a_{4 n} \\
\cdots \cdots
\end{array}
$$

where the numbers in each row form an arithmetic sequence, and the numbers in each column form a geometric sequence, with all common ratios being equal. Given $a_{24}=1, a_{42}=\frac{1}{8}, a_{43}=\frac{3}{16}$, find
$$
a_{11}+a_{22}+a_{33}+a_{44}+\cdots+a_{n n} .
$$