Prove the following by using the principle of mathematical induction for all n ∈ N: 1.2 + 2.3 + 3.4 + ..... + n. (n + 1) = [n (n + 1)(n + 2)/3]

Solution:

Let P (n) be the given statement.

i.e., P (n) : 1.2 + 2.3 + 3.4 + ..... + n. (n + 1) = [n (n + 1)(n + 2)/3]

For n = 1,

P (1) :1.2 = [1 (1 + 1)(1 + 2)/3]

2 = 1.2.3/3

2 = 2, which is true.

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

i.e., 1.2 + 2.3 + 3.4 + .... + k. (k + 1) = [k (k +1)(k + 2)]/3 .....(1)

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

Now, we have,

1.2 + 2.3 + 3.4 + ..... + (k + 1) [(k + 1) + 1]

= [1.2 + 2.3 + 3.4 + ..... + k.(k + 1)] + (k + 1)(k + 2)

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

= (k + 1) (k + 2) [k/3 + 1]

= (k + 1) (k + 2) (k + 3) /3

= {(k + 1) [(k + 1) + 1] [(k + 1) + 2]}/3

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 6

Prove the following by using the principle of mathematical induction for all n ∈ N: 1.2 + 2.3 + 3.4 + ..... + n. (n + 1) = [n (n + 1)(n + 2)/3]

Summary:

We have proved that 1.2 + 2.3 + 3.4 + ..... + n. (n + 1) = [n (n + 1)(n + 2)/3] by using the principle of mathematical induction for all n ∈ N.