Prove the following by the principle of mathematical induction:

i.e., the sum of first n odd natural numbers is n2.

Let P(n): 1 + 3 + 5 + … + (2n - 1) = n2


Let check P(n) is true for n = 1


P(1) = 1 =12


1 = 1


P(n) is true for n = 1


Now, Let’s check P(n) is true for n = k


P(k) = 1 + 3 + 5 + … + (2k - 1) = k2 - - - (1)


We have to show that


1 + 3 + 5 + … + (2k - 1) + 2(k + 1) - 1 = (k + 1)2


Now,


= 1 + 3 + 5 + … + (2k - 1) + 2(k + 1) - 1


= k2 + (2k + 1)


= k2 + 2k + 1


= (k + 1)2


Therefore, P(n) is true for n =k + 1


Hence, P(n) is true for all n∈N.


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