Prove the following by using the principle of mathematical induction for all n ∈ N :
1 + 2 + 3 + .... + n < 1/8 (2n + 1)²

Solution:

Let P (n) be the given statement.

i.e., P (n) : 1 + 2 + 3 + .... + n < 1/8 (2n + 1)²

We note that P (n) is true for n =1,

Since,

P (1) : 1 < 1/8 (2.1 + 1)²

1 < 9/8, which is true.

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

i.e., P (k) : 1 + 2 + 3 + .... + k < 1/8 (2k + 1)² .... (1)

We will now prove that

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

Now, we have

1 + 2 + 3 + ..... + k < 1/8 (2k + 1)²

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

< 1/8 (2k + 1)² + (k + 1) ....[from (1)]

< 1/8 [(2k + 1)² + 8(k + 1) ]

< 1/8 [4k² + 4k + 1+ 8k + 8]

< 1/8 [4k² + 12k + 9]

< 1/8 [2k + 3]²

< 1/8 [2 (k + 1) + 1]²

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 18

Prove the following by using the principle of mathematical induction for all n ∈ N : 1 + 2 + 3 + .... + n < 1/8 (2n + 1)²

Summary:

We have proved that 1 + 2 + 3 + .... + n < 1/8 (2n + 1)² by using the principle of mathematical induction for all n ∈ N.