Asymptotics II

• Use arithmetic/geometric sum formulas
  • Arithmetic:
    • $S = n \cdot \frac{a_{1} + a_{n}}{2}$
    • $S = \frac{n}{2} \cdot [2a_{1} + (n - 1) \cdot d] = n \cdot a_{1} + \frac{(n - 1) \cdot n \cdot d}{2}$
  • Geometric:
    • $S = \frac{a_{1} \cdot (1 - r^{n})}{1 - r}$
    • Infinite ($r < 1$): $S = \frac{a_{1}}{1 - r}$
• $\text{lg}(N) = \log_2(N)$
• $\lceil f(N) \rceil \in \Theta(f(N))$
• $\lfloor f(N) \rfloor \in \Theta(f(N))$

Merge Sort