Q. 17

Prove the following by the principle of mathematical induction:

52n + 2 – 24n – 25 is divisible by 576 for all n ϵ N.

Answer :

Let P(n): 52n + 2 – 24n – 25


For n = 1


= 52.1 + 2 - 24.1 - 25


= 625 – 49


= 576


Since, it is divisible by 576


Let P(n) is true for n=k, so


= 52k + 2 – 24k – 25 is divisible by 576


= 52k + 2 – 24k – 25 = 576λ - - - - - (1)


We have to show that,


= 52k + 4 – 24(k + 1) – 25 is divisible by 576


= 5(2k + 2) + 2 – 24(k + 1) – 25 = 576μ


Now,


= 5(2k + 2) + 2 – 24(k + 1) – 25


= 5(2k + 2).52 – 24k – 24– 25


= (576λ + 24k + 25)25 – 24k– 49 using equation (1)


= 25. 576λ + 576k + 576


= 576(25λ + k + 1)


= 576μ


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


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

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