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Question
Prove the statement by using the Principle of Mathematical Induction:
23n – 1 is divisible by 7, for all natural numbers n.
Solution
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) = `2^(3("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 by 7, for all natural numbers n.
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