Using the principle of mathematical induction, prove each of the following for all n ϵ N:

12 + 32 + 52 + 72 + … + (2n – 1)2 =


To Prove:


12 + 32 + 52 + 72 + … + (2n – 1)2 =


Steps to prove by mathematical induction:


Let P(n) be a statement involving the natural number n such that


(i) P(1) is true


(ii) P(k + 1) is true, whenever P(k) is true


Then P(n) is true for all n ϵ N


Therefore,


Let P(n): 12 + 32 + 52 + 72 + … + (2n – 1)2 =


Step 1:


P(1) = = 1


Therefore, P(1) is true


Step 2:


Let P(k) is true Then,


P(k): 12 + 32 + 52 + 72 + … + (2k – 1)2 =


Now,


12 + 32 + 52 + 72 + … + (2(k + 1)–1)2 =


=


=


=


=


= (Splitting the middle term)


=


= P(k + 1)


Hence, P(k + 1) is true whenever P(k) is true


Hence, by the principle of mathematical induction, we have


12 + 32 + 52 + 72 + … + (2n – 1)2 = for all n ϵ N


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