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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