Suppose we have the following recurrence relation:

then:

Each iteration, the recurrence is replaced with its value as established by the original recurrence relation. Now that we've done a few iterations, let's simplify and see if there is a recognizable pattern.

There definitely seems to be a pattern here. Each iteration we're adding a term; where is the number of iterations that we have made. Now the question is: When is this going to stop?

From the original problem we know that: . We can say that:

Now we can write our simplified equation in terms of .

Now and we know that . Also any constant is . So we can re-write our equation as:

So, the time complexity of this recurrence relations is .