Given
n3 – 7n + 3 is divisible by 3 ∀ n ∈ N
Proof
We will prove the given statement by induction
STEP 1
n = 1
n3 – 7n + 3 = 13 – 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
k3 – 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 = k3 + 1 + 3k2 + 3k – 7k – 7 + 3
= ( k3 – 7k + 3 ) + 3k2 + 3k – 6
= 3x + 3(k2 + k – 2)
= 3 * ( x + k2 + k – 2 )
We can say that (k + 1)3 – 7(k + 1) + 3 is divisible by 3 if k3 – 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.