Answer :

Let P(n): 32n + 2 – 8n – 9


For n = 1


= 32.1 + 2 - 8.1 - 9


= 81 – 17


= 64


Since, it is divisible by 8


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


= 32k + 2 – 8k – 9 is divisible by 8


= 32k + 2 – 8k – 9 = 8λ - - - - - (1)


We have to show that,


= 32k + 4 – 8(k + 1) – 9 is divisible by 8


= 3(2k + 2) + 2 – 8(k + 1) – 9 = 8μ


Now,


= 32(k + 1).32 – 8(k + 1) – 9


= (8λ + 8k + 9)9 – 8k – 8 – 9


= 72λ + 72k + 81 - 8k - 17 using equation (1)


= 72λ + 64k + 64


= 8(9λ + 8k + 8)


= 8μ


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


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


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