Date: Thu, 15 Nov 2001 12:38:04 -0600 (CST) From: Mark Jacobson To: 810-080-01@uni.edu, 810-080-02@uni.edu Subject: Inductive proof group exercises, VERY HELPFUL... Hi 080 students, The URL http://www.cns.uni.edu/~jacobson/c080.html has two new links. The group exercises we did yesterday have been written up and you can look at the solutions. They should help you to do the current homework problems. k+1 k k k k Note that 2 = 2(2) = (1+1)(2) = 2 + 2 Note that (k + 1)! = (k+1)(k!) = k(k!) + 1(k!) = k(k!) + k! You have to manipulate the numbers in many ways and look at the various forms. Its like fishing, in that you have to wait for a bite, i.e. wait for a fish to bite on your bait and hook. The scratch paper and ink are the baiting of the hook and the trying to cast out to different locations looking for the most promising angles to solving the problem as an angler. n ----- \ \ i-1 0 1 2 (n-1)-1 n-1 / 2 = 2 + 2 + 2 + ... + 2 + 2 / ----- n-2 n-1 i = 1 = 1 + 2 + 4 + ... + 2 + 2 Summation notation is just shorthand way of expressing the same idea we have been seeing over and over again. n ----- \ \ n(n+1) / i = 1 + 2 + 3 + 4 + ... + n = -------- / 2 ------ i=1 Mark http://www.cns.uni.edu/~jacobson/080/divisibleBy3.jpg http://www.cns.uni.edu/~jacobson/080/inequalityInduction.jpg linked from: http://www.cns.uni.edu/~jacobson/c080.html