Hidden Geometric Series

$x + xy + xy^2 + xy^3 + \dots = 4$ The left side is the sum of an infinite geometric series with first term as $x$ and common ratio as $y$ .

Since $|y| < 1$ , the sum exists as $S = \frac{a_1}{1-r} = \frac{x}{1-y} = 4 \implies 4 - 4y = x$ ;

Similarly: $y + yx + yx^2 + yx^3 +\dots = 7$ is another infinite geometric series with the first term as $y$ and common ratio as $x$ . Since $|x|<1$ , the sum of the the left side of this series exists and it is equal to $7$ .

Thus, $S' = \frac{a'_1}{1-r'} = \frac{y}{1-x} = 7\implies 7-7x = y$ ;

we have two equations with two unknowns: $\begin{cases} 7 - 7x = y\\ 4 - 4y = x \end{cases}$

$\implies (x, y) = (\frac{8}{9}, \frac{7}{9}) \implies 81xy = 8\times 7 = 56$