46. Find the sum of the infinite decreasing geometric progression:

$$
1+\frac{1}{4}+\left(\frac{1}{4}\right)^{2}+\left(\frac{1}{4}\right)^{3}+\ldots
$$

From the treatise "On the Sphere and Cylinder".
46. Since $\lim S=\frac{a}{1-q}$, then for $a=1$ and $q=\frac{1}{4}$ we have 6 Ppopyov. sororiiik task. 81
$\lim S=\frac{4}{3}$. Archimedes' proof is interesting: we need to find the sum of the terms of an infinitely decreasing progression $a+b+$ $+c+d+\ldots$, if the common ratio $=\frac{1}{4}$. We have:

$$
\begin{gathered}
(b+c+d+\ldots)+\left(\frac{b}{3}+\frac{c}{3}+\frac{d}{3}+\ldots\right)=\left(b+\frac{b}{3}\right)+ \\
+\left(c+\frac{c}{3}\right)+\left(d+\frac{d}{3}\right)+\ldots=\frac{4}{3} b+\frac{4}{3} c+\frac{4}{3} d+\ldots= \\
=\frac{1}{3}(4 b+4 c+4 d+\ldots)=\frac{1}{3}(a+b+c+d+\ldots)
\end{gathered}
$$

from this

$$
b+c+d+\ldots=\frac{1}{3} a
$$

and

$$
a+b+c+d+\ldots=\frac{4}{3} a
$$