Practice proving theorems by express your argument in a succinct and also logically continuous form.

You are watching: Strong induction vs weak induction

In this page, we will comment on the rule of mathematicalinduction and prove nature of numbers usinginduction.

This lecture coincides to ar 2.3 that Ensley and Crawley"s book.

## Mathematical Induction: basic Principle

Mathematical induction is a very common an approach for provingproperties of herbal numbers (and other discrete frameworks such assets, relations and also trees that we will certainly study very soon).

Let us imagine one infinitely lengthy sequence of tiles arranged in astraight heat (close sufficient to every other), and also let us tip dominonumber 1.

We wish to argue that every domino will certainly fall. Right here is just how we deserve to argue:

Base case: The first domino drops (we kicked it, so it falls).

Inductive Step: whenever a domino numbered drops (or every the dominos numbdered 1 come fall), then its follower numbered , additionally falls.

Therefore, us conclude that all the dominoes will certainly fall.

The argument over is the crux of induction. Come prove a residential property over all organic numbers , we might argue as follows:

The residential or commercial property is true because that (the first natural number).

If the building is true for some organic number (or if the property is true because that all organic numbers as much as ), then it is true for organic number .

## Weak Induction Proofs

Weak induction supplies a simple method for prove a building because that all natural numbers . I.e, .

Proof by weak induction proceeds in basic three steps!

Step 1: check the base case. Verify that holds.

Step 2: compose down the Induction Hypothesis, which is in the form. (All you need to do is to number out what and also are!)

Step 3: Prove the Induction Hypothesis (that you composed down).This step usually provides use of the meaning of the recursion and also the premise .

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### Example 1

Induction deserve to be really helpful to guess and also prove closed forms of sequences. Consider a an easy one:

We have actually .Therefore right here is our guess: . Just how do us prove it?

Claim: .

Proof by induction

Base Case: us verify the . Therefore this works.

Induction Hypothesis: For every , If then .

Proof the induction hypothesis:

Let be any type of given natural number such that .

We seek to prove that . In fact, &=& a_n + 2n + 1 & &=& n^2 + 2n + 1 & < a_n = n^2, mboxby induction hypothesis> &=& (n+1)^2 endarray " style="vertical-align: -67px" />