Can not figure out complexity of this recurrence

Question

I am refreshing on Master Theorem a bit and I am trying to figure out the running time of an algorithm that solves a problem of size n by recursively solving 2 subp

Answer 1

May be you could think of it this way

when

n = 1, T(1) = 1
n = 2, T(2) = 2
n = 3, T(3) = 4
n = 4, T(4) = 8
n = 5, T(5) = 16

It is easy to see that this is a geometric series 1 + 2+ 4+ 8 + 16..., the sum of which is first term (ratio^n - 1)/(ratio - 1). For this series it is

1 * (2^n - 1)/(2 - 1) = 2^n - 1.

The dominating term here is 2^n, therefore the function belongs to Theta(2^n). You could verify it by doing a lim(n->inf) [2^n / (2^n - 1)] = +ve constant.

Therefore the function belongs to Big Theta (2^n)