% These slides are designed for 52 x 19 terminals
% but should work with slightly larger or smaller
% terminals. Should be OK with unix "less" and
% screens with fewer than 30 lines. Issue /%%etc.
% Start at lines 10, 40, 70, 100, etc.; use
% vi "set number" and "set nonumber"
% line 10: s1%%etc., line 40: s2%%etc., etc.
% unix, vi, less -- old school!
s1%%%
Today: Finish Fibonacci
- Matrix Mult Idea
- Exact formula
Review: Induction
s2%%%
Principle of Induction:
Let P(1),P(2),P(3),... be sequence
of true/false values.
If
- P(1) is true, and
- for any integer k > 0 we have
P(k) => P(k+1)
then
P(1)= true = P(2) = P(3) = ...
s3%%%
Variant of Principle of Induction:
Let P(1),P(2),P(3),... be sequence
of true/false values.
If
- P(1) is true, and
- for any integer k > 0 we have
P(1) & P(2) & ... & P(k)
=> P(k+1)
then
P(1)= true = P(2) = P(3) = ...
s3%%%
Another Variant
Let S be a subset of N = {1,2,3,...}.
Assume that 1 lies in S, and that for
any integer k, if k in S then
k+1 in S.
Then S = N.
s3%%%
Be careful: with Fibonacci numbers, you
typically have to treat cases involving only
f(1) and f(2) a bit differently.