Hello I am working on solving a relation of the form to find time complexity of an algorithm:
(1) $T(n) = 2T(\lfloor(\frac{n}{2})\rfloor +17) + n$
This is a homework problem and I am required to use the substitution method. I understand the substitution/induction part of the proof but I am struggling to understand how to form a proper base case.
We have to show that a base case for $n = 34$ holds true for all $ n <= 34 $.
I've seen other examples of problems that look similar such as:
(2) $T(n) = 2T(\lfloor(\frac{n}{2})\rfloor) + n$
Where they use $n = 0$ as a base case, but this doesn't make any sense because what would be the use of running an algorithm on a data set of size 0?
I think it would help if someone could explain how to get a base case for a simpler problem such as equation 2, and then how the process changes for an equation like equation 1.
Thank You