A probability problem by Victor Paes Plinio

$a$ and $b$ are positive integers. If $a$ and $b$ are two consecutive terms of a geometric progression of common ratio $\frac{1}{2}$ and the independent term of ${\left(ax-\frac{b}{\sqrt{x}}\right)}^{12}$ is $7920$ , find the value of $a+b$ .

Problem from ITA 2018 (Instituto Tecnológico de Aeronáutica)