Question

For any natural number n, 7ⁿ – 2ⁿ is divisible by 5.

Answer 1

Let P(n) : 7ⁿ – 2ⁿ 

Step 1 : 

P(1) : 7¹ – 2¹ = 5λ which is divisible by 5. So it is true for P(1). 

Step 2 : 

P(k): 7^k – 2^k = 5λ. Let it be true for P(k) 

Step 3 : 

P(k + 1) = 7^{k + 1} – 2^{k + 1}

So, it is true for P(k + 1) 

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