Question

For every positive integer n,P.T 7^n-3^n is divisible by 4

Answer 1

Let P(n) : (7n – 3n) is divisible by 4.

For n = 1, the given expression becomes (7 1 - 3 1) = 4, which is divisible by 4.

So, the given statement is true for n = 1, i.e., P(1) is true.

Let P(k) be true. Then,

P(k): (7k - 3k) is divisible by 4.

⇒ (7k - 3k) = 4m for some natural number m.

Now, {7(k + 1) - 3 (k + 1)} = 7(k + 1) – 7 ∙ 3k + 7 ∙ 3k - 3 (k + 1)
(on subtracting and adding 7 ∙ 3k)

= 7(7k - 3k) + 3 k (7 - 3)

= (7 × 4m) + 4 ∙ 3k

= 4(7m + 3k), which is clearly divisible by 4.

∴ P(k + 1): {7(k + 1) - 3 (k + 1)} is divisible by 4.

⇒ P(k + 1) is true, whenever P(k) is true.

Hence, by the principle of mathematical induction, P(n) is true for all n ∈ N.

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