If then use the principle of mathematical induction to show that for every positive integer n.

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.








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 every positive integer n.


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