How to do Mathematical Induction, Easy guide for beginners

What is mathematical induction?

It is a mathematical way to prove things by deductive reasoning.

An example for deductive reasoning

4 is divisible by 2
Any number divisible by two is an even number,therefore
4 is an even number. 

You should have got the idea by now.

Principle of mathematical induction

A popular example is a set of tiles placed near each other

The first tile falls,
If any tile falls its successor falls. 

Mathematical induction is very easy, Make sure you write each step and solve it by yourself.

How to solve Mathematical induction. 

Here is an example for Mathematical induction, The formula to find the sum of positive integers 1,2,3...n is given below.

We can prove the equation by giving different values for n.  But we cannot check the value for all integers.  We need something like a chain reaction for that.  If the value is true for a positive integer k, then (k+1) should also be true in the equation. 

Solving a Mathematical induction problem

Here's a pattern of numbers, note that:

The sum of first two odd natural numbers is the square of second natural number  sum of first three odd natural numbers is the square of third natural number, the pattern goes on

We need a formula for this patternz It is given below

To solve the Mathematical induction

Let us write P(n)

 

When putting the value of P(n)=1, we get 1 which is true.

Let P(k) be true 

Now assume P(k+1) is also true. Get the value for P(k+1) 

Change the value of P(k) from equation to P(k+1)

Look at the highlighted part its the same value as P(k)

We replace it with RHS value of P(k)

 

Equate the LHS, we get a Quadratic equation.  Solve it further, we get the same value 

Hence proved.

Another example of mathematical induction. 

For all n>=1 (Greater than or equal to), prove that

Step 1:

Let P(n) be

Step 2:

For n=1

Step 3:

Now, n=k

Step 4:

Now n=(k+1)

Step 5:

Replacing with value of P(k)

Step 6:

Equating

 we get the same value of RHS from Step 4.

That was so easy! Any questions? Comment below.