# Using the princip

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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