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).