Problems on Induction

Mathematical Induction

DescriptionDescription

Three StepsThree Steps

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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 onPreliminaries/Background and Preview/Induction by M. Seppälä

Mathematical Induction

Problem 1Problem 1

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1.

2

n

k

n nkShow that for all positive integers n,

SolutionSolution

Solved Problems onPreliminaries/Background and Preview/Induction by M. Seppälä

Mathematical InductionShow that for all positive integers n,

SolutionSolution

Solved Problems onPreliminaries/Background and Preview/Induction by M. Seppälä

2

1 12

1

1 1 .2 2

mn m

n

m mn

Problem 2Problem 2

SumsShow that for all positive integers n,

SolutionSolution

Problems onPreliminaries/Background and Preview/Induction by M. Seppälä

2 3 3 31 2 3 1 2 .n n

Problem 3Problem 3

Mathematical InductionFind 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+1 must be of the same color. Hence the car c1

is 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.

SolutionSolution

Problems onPreliminaries/Background and Preview/Induction by M. Seppälä

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Problem 4Problem 4

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.

DefinitionDefinition

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

1 51.618033989... .

2

Problems onPreliminaries/Background and Preview/Induction by M. Seppälä

Problem 5Problem 5

Let be the negative solution of the equation x2 = 1 + x. 1 5

2

Show that

.n n

nF