Problems on Induction

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Problems on Induction. Mathematical Induction. Description.

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### Problems on Induction

Mathematical Induction

Description

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.

Three Steps

Show that the statement is true for the first value of the parameter.

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2

Induction Assumption: The statement holds for some value m of the parameter.

Show that the statement is true for the parameter value m+1.

3

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

Problem 2

Show that for all positive integers n,

Solution

Solved Problems on

Preliminaries/Background and Preview/Induction by M. Seppälä

Sums

Problem 3

Show that for all positive integers n,

Solution

Problems on

Preliminaries/Background and Preview/Induction by M. Seppälä

Mathematical Induction

Problem 4

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={c1 , c2 , c3 ,…, cn+1} be a set of n+1 cars. Then the cars c1 , c2 , c3 ,…, cn form a set of n cars. By the induction Assumption (2) they must be of the same color. Likewise the cars c2 , c3 ,…, cn+1must be of the same color. Hence the car c1is of the same color as the car c2, and the car c2 is of the same color as the car c3. And so on. Consequently all cars are of the same color.

1

2

3

Solution

Problems on

Preliminaries/Background and Preview/Induction by M. Seppälä

Fibonacci Numbers – Challenge

The Fibonacci Numbers Fn , n = 0,1,2,… are defined by setting F0 = 0, F1= 1, and Fn + 1 = Fn + Fn – 1 for n > 1.

Definition

The positive solution α to x2 = 1 + x is the Golden Ratio. We have

Let be the negative solution of the equation x2 = 1 + x.

Show that

Problem 5

Problems on

Preliminaries/Background and Preview/Induction by M. Seppälä