Answer :

Given.


We need to prove that using the principle of mathematical induction.


Step 1: When n = 1, we have




Hence, the equation is true for n = 1.


Step 2: Let us assume the equation true for some n = k, where k is a positive integer.



To prove the given equation using mathematical induction, we have to show that.


We know Ak+1 = Ak × A.





However, we have i2 = –1






Hence, the equation is true for n = k + 1 under the assumption that it is true for n = k.


Therefore, by the principle of mathematical induction, the equation is true for all positive integer values of n.


Thus, for all n ϵ N.


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