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23n – 1 is divisible by7,

Question:

23n – 1 is divisible by7, for all natural numbers n.

Solution:

According to the question,

P(n) = 23n – 1 is divisible by 7.

So, substituting different values for n, we get,

P(0) = 20 – 1 = 0 which is divisible by 7.

P(1) = 23 – 1 = 7 which is divisible by 7.

P(2) = 26 – 1 = 63 which is divisible by 7.

P(3) = 29 – 1 = 512 which is divisible by 7.

Let P(k) = 23k – 1 be divisible by 7

So, we get,

⇒ 23k – 1 = 7x.

Now, we also get that,

⇒  P(k+1) = 23(k+1) – 1

= 23(7x + 1) – 1

= 56x + 7

= 7(8x + 1) is divisible by 7.

⇒ P(k+1) is true when P(k) is true.

Therefore, by Mathematical Induction,

P(n) = 23n – 1 is divisible by7, for all natural numbers n.

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