Prove the following by the principle of mathematical induction:

52n – 1 is divisible by 24 for all n ϵ N

Let P(n): 52n - 1 is divisible by 24


Let’s check For n =1


P(1): 52 - 1 = 25 - 1


= 24


Since, it is divisible by 24


So, P(n) is true for n=1


Now, for n=k


52k - 1 is divisible by 24


P(k): 52k - 1 = 24λ - - - - - - - (1)


We have to show that,


52k + 1 - 1 is divisible by 24


52(k + 1) - 1 = 24μ


Now,


52(k + 1) - 1


= 52k.52 - 1


= 25.52k - 1


= 25.(24λ + 1) - 1 using equation (1)


= 25.24λ + 24


= 24λ


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


Hence, P(n) is true for all nN by PMI


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