Question

1. Prove that
$2 {}^{2n} - 1$
is divisible by 3 by using PMI.

Please give proper answer with space between each line to understand easier.

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Answer 1

Answer:-

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) : 22(k+1) – 1 = 2 2k + 2 – 1 = 22k . 22 – 1 = 2 2k . 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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