Problems on Induction

Mathematical Induction Description Three Steps Mathematical Induction applies to statements which depend on a parameter that takes typically integer values starting from some initial value. It can be seen as a machine that produces a proof of a statement for any finite value of the parameter in question. Show that the statement is true for the first value of the parameter. Induction Assumption: The statement holds for some value m of the parameter. Show that the statement is true for the parameter value m+1. Solved Problems on Preliminaries/Background and Preview/Induction by M. Seppälä

Mathematical Induction Problem 1 Show that for all positive integers n, Solution Solved Problems on Preliminaries/Background and Preview/Induction by M. Seppälä

Mathematical Induction Show that for all positive integers n, Solution Solved Problems on Preliminaries/Background and Preview/Induction by M. Seppälä Problem 2

Sums Show that for all positive integers n, Solution Problems on Preliminaries/Background and Preview/Induction by M. Seppälä Problem 3

Mathematical Induction Find the error in the following argument pretending to show that all cars are of the same color. This is a modification of an example of Pólya. If there is only one car, all cars are of the same color. Assume that all sets of n cars are of the same color. Let S={c 1, c 2, c 3,…, c n+1 } be a set of n+1 cars. Then the cars c 1, c 2, c 3,…, c n form a set of n cars. By the induction Assumption (2) they must be of the same color. Likewise the cars c 2, c 3,…, c n+1 must be of the same color. Hence the car c 1 is of the same color as the car c 2, and the car c 2 is of the same color as the car c 3. And so on. Consequently all cars are of the same color. Solution Problems on Preliminaries/Background and Preview/Induction by M. Seppälä Problem 4

Fibonacci Numbers – Challenge The Fibonacci Numbers F n, n = 0,1,2,… are defined by setting F 0 = 0, F 1 = 1, and F n + 1 = F n + F n – 1 for n > 1. Definition The positive solution α to x 2 = 1 + x is the Golden Ratio. We have Problems on Preliminaries/Background and Preview/Induction by M. Seppälä Problem 5 Let be the negative solution of the equation x 2 = 1 + x. Show that