Lecture 19: Sequences, Series and Such + intro to counting

Last time we talked about sequences.  Sequences such as:

3, -9, 27, -81, 243, ...

Mostly sequences we worry about are arithmetic or geometric, meaning
you can get from one number to the next by adding or multiplying the
same number.  When presented with a sequence, one approach to figuring
out what to do is to:

subtract consecutive elements, or
divide consecutive elements.  If these are the same, or close to the
same, then you have a good clue of what is happening:

For example, from the above, we get:

subtracting  -12, 36, 108, 324...., wow that's giving nothing!
dividing:    -3, -3, -3, -3, -3... cool!

so we can immediately write a recursive formula:

a0 = 3
an = -3an-1.

But it is really nice to have an explicit formula.

A really really good method for finding explicit formulas is "guess
and check".  Honestly.  Works pretty often, especially after you have
some practice.  But there is also an formulaic way to go from a
recursive formula to an explicit formula.  This is especially useful
if you have a recursive formula that has multiple terms (like the
fibonacci sequence).

See Chapter 7.2 (in the new book) under "linear recurrance relations"

oh, and we did the beginning of counting also.

See, for example, berkeley counting notes