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

Solution:

Let P (n) be the given statement.

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

For n = 1,

P (1) : 1 = (2 x 1)/(1 + 1) = 2/2 = 1, which is true.

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

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

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

Now, we have,

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

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

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

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

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

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

= 2 (k² + 2k + 1)/(k + 1)(k + 2)

= 2 (k + 1)²/(k + 1)(k + 2)

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

= 2 (k + 1) / [(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 3

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

Summary:

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