% 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.