Day 10: mat385

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Today:

• Announcements:
  • Reminder: turn off your rectangles.

  • Your 3.1 homework is returned (graded 1, 24, and 42).
    • #1: no problems!

    • #24: These "Second Principle" problems require two base cases. The reason is because you're going to invoke the process on the two prior cases ($P(k)$ and $P(k-1)$).

      Some of you proved the result directly -- which is okay, except the problem asked you to prove it using the Second Principle....

    • #42: This involved an inequality, which always makes things a little trickier.

      You do need to justify, however: you can't just say $\left(\frac{5}{2}\right)^{k}+\left(\frac{5}{2}\right)^{k-1} \le \left(\frac{5}{2}\right)^{k+1}$

      Perhaps the best strategy is to force a $\left(\frac{5}{2}\right)^{k+1}$ to pop out of the left side above:

      \[ \left(\frac{5}{2}\right)^{k+1}\left[\left(\frac{2}{5}\right)+\left(\frac{2}{5}\right)^{2} \right] = \left(\frac{5}{2}\right)^{k+1} \left(\frac{24}{25}\right) \le \left(\frac{5}{2}\right)^{k+1} \]

• Today:
  • We continue counting things, and so we finish off section 3.3: Analysis of Algorithms