Question

Prove by induction that n³-n is a multiple of 3.

Answer 1

Let, P(n) be the statement, P(n) : n³ - n is a multiple of 3 for all natural number n.

Basic Step : P(1) : 1-1 = 0, is divisible by 3.

So, P(1) holds.

Induction hypothesis : let, P(n) be true for some natural number m.

i.e. P(m) : m³ - m is divisible by 3, so, m³ - m = 3k for some integer k.

Inductive step :

Now, P(m+1) : (m+1)³ - (m+1) = m³ + 3m² + 3m + 1 - m - 1 = (m³ - m) + 3m(m+1) = 3k + 3m(m+1) = 3(k+m²+m) , divisible by 3.

So, P(m+1) holds true whenever P(m) holds true.

So, P(n) is true for all natural number n.

i.e. n³ - n is divisible by 3 for all natural number n.