Prove the following by using the principle of mathematical induction for all n ∈ N :
1/(1.4) + 1/(4.7) + 1/(7.10) + .... + 1/[(3n - 2)(3n + 1)] = n/(3n + 1)

Solution:

Let P (n) be the given statement.

i.e., P (n) : 1/(1.4) + 1/(4.7) + 1/(7.10) + .... + 1/[(3n - 2)(3n + 1)] = n/(3n + 1)

For n = 1,

P (1) : 1/1.4 = 1/(3.1 + 1)

1/4 = 1/(3.1 + 1)

1/4 = 1/4, which is true.

Assume that P (k) is true for some positive integer k.

i.e., P (k) : 1/(1.4) + 1/(4.7) + 1/(7.10) + .... + 1/[(3k - 2)(3k + 1)] = k/(3k + 1) ....(1)

We will now prove that P (k + 1) is also true.

Now, we have

1/(1.4) + 1/(4.7) + 1/(7.10) + .... + 1/[(3(k + 1) - 2][3(k + 1) + 1]

= 1/(1.4) + 1/(4.7) + 1/(7.10) + .... + 1/[(3k - 2)(3k + 1)] + 1/[(3k + 1)(3k + 4)]

= k/(3k + 1) + 1/[(3k + 1)(3k + 4)] ....[from (1)]

= k (3k + 4)/[(3k + 1)(3k + 4)]  + 1/[(3k + 1)(3k + 4)] 

= [3k² + 4k + 1] / [(3k + 1)(3k + 4)]

= [3k² + 3k + k + 1] / [(3k + 1)(3k + 4)]

= [3k (k + 1) + (k + 1)] / [(3k + 1)(3k + 4)]

= [(3k + 1)(k + 1)] / [(3k + 1)(3k + 4)]

= (k + 1) / (3k + 4)

= (k + 1)/[3(k + 1) + 1]

Thus P (k + 1) is true, whenever P (k) is true.

Hence, from the principle of mathematical induction, the statement P (n) is true for all natural numbers i.e., n ∈ N .

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NCERT Solutions Class 11 Maths Chapter 4 Exercise 4.1 Question 16

Prove the following by using the principle of mathematical induction for all n ∈ N : 1/(1.4) + 1/(4.7) + 1/(7.10) + .... + 1/[(3n - 2)(3n + 1)] = n/(3n + 1)

Summary:

We have proved that 1/(1.4) + 1/(4.7) + 1/(7.10) + .... + 1/[(3n - 2)(3n + 1)] = n/(3n + 1) by using the principle of mathematical induction for all n ∈ N