Finding a Big-O notation of: $\sum\limits_{i=1}^{k} ( t(a_i n)) + n$

Question

I'm trying to find a Big-O notation of:

$\displaystyle\sum_{i=1}^{k} ( t(a_in)) + n$, where $\displaystyle\sum_{i=1}^{k} (a_i) < 1$

using a recursion tree method and substitution method. I've spent many hours trying to break this problem down, but it seems very complex (hence the summation of recurrences).

Any help would be great!

Thanks.

Answer 1

Hint: Presumably you are taking the floor of $a_in$ and have $$t(n)=\sum_{i=1}^{k} t(\lfloor a_in\rfloor) + n$$ Can you argue that $t$ is increasing so $$\sum_{i=1}^{k} t(\lfloor a_in\rfloor) \le t(n-1)$$ at least eventually?