Question

Use induction to prove that n³ – 7n + 3, is divisible by 3, for all natural numbers n.

Answer 1

Let P(n) : n³ – 7n + 3 

Step 1:

P(1) = (1)³ – 7(1) + 3 

= 1 – 7 + 3 = -3 which is divisible by 3 

So, it is true for P(1).

Step 2: 

P(k) : k³ – 7k + 3 = 3λ. Let it be true 

⇒ k³ = 3λ + 7k – 3

Step 3: 

P(k + 1) = (k + 1)³ – 7(k + 1) + 3 

= k³ + 1 + 3k² + 3k – 7k – 7 + 3 

= k³ + 3k² – 4k – 3 

= (3λ + 7k – 3) + 3k² – 4k – 3 (from Step 2) 

= 3k² + 3k + 3λ – 6 

= 3(k² + k + λ – 2) which is divisible by 3. 

So it is true for P(k + 1). 

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