PRINCIPLE OF MATHEMATICAL INDUCTION EXAMPLES

About "Principle of Mathematical Induction Examples"

Principle of Mathematical Induction Examples

Here we are going to see some mathematical induction problems with solutions.

Define mathematical induction :

Mathematical Induction is a method or technique of proving mathematical results or theorems

The process of induction involves the following steps.

Principle of Mathematical Induction Examples

Question 1 :

By the principle of mathematical induction, prove that, for n ≥ 1

1.2 + 2.3 + 3.4 + · · · + n.(n + 1) = n(n + 1)(n + 2)/3

Solution :

Let p(n) =  1.2 + 2.3 + 3.4 + · · · + n.(n + 1) = n(n + 1)(n + 2)/3

Step 1 :

put n = 1

p(1)  =  1.2 + 2.3 + 3.4 + · · · + 1.(1 + 1) = 1(1 + 1)(1 + 2)/3

  1  =  1

Hence p(1) is true.

Step 2 :

Let us assume that the statement is true for n = k

p(k) = 1.2 + 2.3 + · · · + k.(k + 1) = k(k + 1)(k + 2)/3  ----(1)  

We need to show that P(k + 1) is true. Consider,

Step 3 :

Let us assume that the statement is true for n = k + 1

p(k+1) 

 1.2 + 2.3 + · · · k(k + 1) + (k+1).(k + 2) = (k+1)(k+2)(k+3)/3

By applying (1) in this step, we get

Hence, by the principle of mathematical induction for n ≥ 1

1.2 + 2.3 + 3.4 + · · · + n.(n + 1) = n(n + 1)(n + 2)/3

Question 2 :

Using the Mathematical induction, show that for any natural number n ≥ 2,

(1 − 1/2²) (1 − 1/3²)(1 − 1/4²) ...............(1 − 1/n²) =(n + 1)/2n

Solution :

Let p(n)  =  (1 − 1/2²) (1 − 1/3²)(1 − 1/4²) ...............(1 − 1/n²)

Step 1 :

(1 − 1/2²) (1 − 1/3²)(1 − 1/4²) ...............(1 − 1/n²) =(n + 1)/2n

put n = 2

p(2)  =  (3/4) =  3/4

Hence p(2) is true.

Step 2 :

Let us assume that the statement is true for n = k

(1 − 1/2²) (1 − 1/3²) ..................(1 − 1/k²) =(k + 1)/2k  --(1)  

We need to show that P(k + 1) is true. Consider,

Step 3 :

Let us assume that the statement is true for n = k + 1

p(k+1) 

(1 − 1/2²) (1 − 1/3²) ................(1 − 1/k²) + (1 − 1/(k+1)²)

=  (k + 2)/2(k + 1)

By applying (1) in this step, we get

(k + 2)/2(k + 1)  =  (k + 2)/2(k + 1)

Hence, by the principle of mathematical induction natural number n ≥ 2,

(1 − 1/2²) (1 − 1/3²)(1 − 1/4²) ...............(1 − 1/n²) =(n + 1)/2n

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