Solve $T(n) = T( \lfloor 2n / 10\rfloor ) + \Theta(n^3)$ recurrence

Question

I don't know how to solve the following recurrence. Particularly, I don't know how to cope with the $2n$ term: $$T(n) = T( \lfloor 2n / 10\rfloor ) + \Theta(n^3)$$

Master theorem of recurrences doesn't really explains how to... I was thinking that this would solve it:

$$T(n) = 2T(\lfloor n / 10\rfloor) + \Theta(n^3)$$

But I'm not sure. How do I approach this?

Answer 1

\begin{align}T(n) &= T(\lfloor2n/10 \rfloor) + \Theta(n^3) \\ &= T(\lfloor n/5\rfloor) + \Theta(n^3) &(1)\\ &=T(n/5) + \Theta(n^3) & (2)\\ &= \Theta(n^3) & (3)\end{align}

I have applied:

$(1)$ Fraction simplification: $2n/10 = n/5$.

$(2)$ Removing floors. In general you can remove all floors and ceilings in recurrences that may be solved with the master theorem, that is, recurrences of the form $$T(n) = a T(n/b) + g(n)$$ (this is a proved fact).

$(3)$ Application of the master theorem.