Solution:

The answer is $(4, -2)$ and $\left(\frac{5}{2}, -5\right)$.

Since $10, a, b$ is an arithmetic progression, we have
$$a = \frac{1}{2}(10 + b).$$
Also, we have $a + a b = 2 b$, and so
$$a(1 + b) = 2b.$$
Substituting the expression for $a$ gives
$$(10 + b)(1 + b) = 4b.$$
Solving this quadratic equation gives the solutions $b = -2$ and $b = -5$. The corresponding values for $a$ can be found by $a = \frac{1}{2}(10 + b)$.