**Given**

n^{3} – 7n + 3 is divisible by 3 ∀ n ∈ N

**Proof**

We will prove the given statement by induction

**STEP 1**

n = 1

n^{3} – 7n + 3 = 1^{3} – 7*1 + 3 = -3

-3 is divisible by 3. Therefore, the statement is true for n = 1

**STEP 2**

Let the given statement be true for n = k

k^{3} – 7k + 3 = 3x

Now, we need to prove that if the statement is true for n = k then it is also true for n = k + 1

(k + 1)^{3} – 7(k + 1) + 3 = k^{3} + 1 + 3k^{2} + 3k – 7k – 7 + 3

= ( k^{3} – 7k + 3 ) + 3k^{2} + 3k – 6

= 3x + 3(k^{2} + k – 2)

= 3 * ( x + k^{2} + k – 2 )

We can say that (k + 1)^{3} – 7(k + 1) + 3 is divisible by 3 if k^{3} – 7k + 3 is divisible by 3.

Therefore, we can say that if the given statement is true for n = k, then it is also true for n = k + 1. Hence, by the principle of mathematical induction, the given statement is true ∀ n ∈ N.