163. Reduced Share. A father gave his children 6 dollars for entertainment, which was to be divided equally. But two young cousins joined the company. The money was divided equally among all the children, so that each child received 25 cents less than originally intended. How many children were there in total?
163. The share of each child decreased by $\frac{2}{24}$ of the entire sum. Further, $24=2 \cdot 12=3 \cdot 8=4 \cdot 6$. From these representations of the number 24, we choose the pair $q_{1} c_{1}=q_{2} c_{2}$ such that $q_{1}+1=q_{2}$, and $c_{1}=c_{2}+2$. Thus, there were initially 6 children, and then there were 8.

$[$ [

$164^{\star}$. Let

$$
\begin{gathered}
S=1+2 x+3 x^{2}+4 x^{3}+\ldots \\
x S=\quad x+2 x^{2}+3 x^{3}+\ldots \\
(1-x) \cdot S=1+x+x^{2}+x^{3}+\cdots=\frac{1}{(1-x)}
\end{gathered}
$$

From this, $S=\frac{1}{(1-x)^{2}}$.