Day 10: mat385

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Today:

Announcements
- 2.2 homework returned: graded #23, 27, 56
  - #23: fairly straightforward 1st principle problem. This is a relatively famous formula, which you should probably retain (or at least recall that it's the formula for the sum of a geometric series).
  - #27: generally one does not work with the entirety of $P(k+1)$, but rather with just one side.
    In this problem, you can actually demonstrate $P(k+1)$ without invoking $P(k)$! That means that you really haven't done an induction proof, but have rather done a direct proof: \[ (k+1)^2 \ge 2(k+1)+3 \] \[ \iff \] \[ k^2+2k+1 \ge 2k+5 \] \[ \iff \] \[ k^2 \ge 4 \]
    So $P(k+1)$ for $k\ge 2$ means $P(n)$ $\forall n\ge 3$
  - #56: this is very much like the proof I did last time that the product of two consecutive integers is even. This time we simply want to show that a product is divisible by 3: that is, the product can be written as $3m$, where $m$ is an integer.
- I did put the notes from last time on-line, and I'll continue that each day.
Today we'll finish section 2.5: Recurrence Relations.
Then we'll start section 2.6: Analysis of Algorithms, if we have any time left.
Links:
- Notes from class on recursion
- Second day on recursion

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