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Question
Prove statement by using the Principle of Mathematical Induction for all n ∈ N, that:
22n – 1 is divisible by 3.
Solution
Let the statement P(n) given as
P(n): 22n – 1 is divisible by 3, for every natural number n.
We observe that P(1) is true.
Since 22 – 1 = 4 – 1 = 3
1 is divisible by 3.
Assume that P(n) is true for some natural number k.
i.e., P(k) : 22k – 1 is divisible by 3.
i.e., 22k – 1 = 3q, where q ∈ N,
Now, to prove that P(k + 1) is true,
We have P(k + 1) : `2^(2(k + 1)) - 1`
= `2^(2k + 2) – 1`
= 22k . 22 – 1
= 22k . 4 – 1
= 3.22k + (22k – 1)
= 3.22k + 3q
= 3(22k + q) = 3m, where m ∈ N,
Thus P(k + 1) is true, whenever P(k) is true.
Hence, by the Principle of Mathematical Induction P(n) is true for all natural numbers n.
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