## Task 21/85

For two real numbers $a$ and $b$, all third real numbers $x_{i}$ are to be determined such that the three products of one of the three numbers and the sum of the other two form an arithmetic sequence of the first order.
With the approach

$$
a\left(b+x_{i}\right)=b\left(a+x_{i}\right)+d=x_{i}(a+b)+2 d
$$

it follows for the difference $d$ of the arithmetic sequence and the sought real numbers $x_{i}$

$$
d=\frac{a^{2} b-a b^{2}}{2 a-b} \quad ; \quad x_{i}=\frac{d}{a-b}=\frac{a b}{2 a-b}
$$