Prove the following by using the principle of mathematical induction for all n ∈ N: 1.2.3 + 2.3.4 + .... + n (n + 1)(n + 2) = n (n + 1)(n + 2)(n + 3)/4

Solution:

Let P (n) be the given statement.

i.e., P (n) = 1.2.3 + 2.3.4 + .... + n (n + 1)(n + 2) = n (n + 1)(n + 2)(n + 3)/4

For n = 1,

P (1) : 1.2.3 = 1 (1 + 1)(1 + 2)(1 + 3)/4

6 = 6 , which is true.

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

i.e., 1.2.3 + 2.3.4 + .... + k (k + 1)(k + 2) = k (k + 1)(k + 2)(k + 3)/4 ....(1)

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

Now, we have,

1.2.3 + 2.3.4 + ..... + (k + 1) [(k + 1) + 1] [(k + 1) + 2]

= [1.2.3 + 2.3.4 + ..... + k (k + 1)(k + 2)] + (k + 1)(k + 2)(k + 3)

= k (k + 1)(k + 2)(k + 3)/4 + (k + 1)(k + 2)(k + 3) .....[from (1)]

= (k + 1)(k + 2)(k + 3) [k/4  + 1]

= (k + 1)(k + 2)(k + 3) [ (k + 4) /4]

= (k + 1)(k + 2)(k + 3) (k + 4) /4 

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 4

Prove the following by using the principle of mathematical induction for all n ∈ N: 1.2.3 + 2.3.4 + .... + n (n + 1)(n + 2) = n (n + 1)(n + 2)(n + 3)/4

Summary:

We have proved that 1.2.3 + 2.3.4 + .... + n (n + 1)(n + 2) = n (n + 1)(n + 2)(n + 3)/4 by using the principle of mathematical induction for all n ∈ N.