Does this equation has a closed form solution?

Question

We have $K$ non-negative coefficients: $a_1,a_2,\dots,a_K,A_1,A_2,\dots,A_K$, where $A_i\geq0,\;a_i\in(0,1),\;\sum A_i<T$. The equation is: $$\sum_{i=1}^K\frac{A_i}{1-a_ix}=T,\quad x\in(0,1)$$