Given a recursive function $T(n)=T(a_1\cdot n)+\dots +T(a_k\cdot n)+\Theta(n)$ such that $\forall a_i: 0<a_i<1$, what is the most general thing I can say about the sum of the cost of the nodes at each level of the recursion tree? I've looked at a few, and for the ones I've looked at it's always come out to $(a_1+\dots +a_k)^i$ where $i$ is the level in the tree.
Could anyone explain this? If it matters it's in the context of trying to prove that $T(n)=\Theta(n) \Leftrightarrow a_1+\dots +a_k < 1$