Q. 104.3( 8 Votes )

Using the princip

Answer :

Let P(n) = 23n – 1,


For n = 1,


P (1) = 23(1) -1


= 8-1


= 7


Which is divisible by 7.


Hence P (1) is true.


Consider P(k) to be true.


i.e. 23(k) -1 is divisible by 7.


23(k) -1 = 7λ for some λ N


We have to prove that P(k+1) is true i.e. 23(k+1) -1 is divisible by 7.


Now,


23(k+1) -1 = 23k × 23 – 1


= (7λ +1) 23 – 1


= 56λ + 8 – 1


= 7(8λ +1)


Which is divisible by 7.


P(k+1) is true when P(k) is true.


Hence by Mathematical induction P(n) is true for all n N.


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