n3 + 5n is divisible by 6 ∀ n ∈ N
We will prove the given statement using the principle of mathematical induction
We need to check if the statement is true for n = 1.
n3 + 5n = 13 + 5(1) = 1 + 5 = 6
Therefore, the statement is true for n = 1.
Let the statement be true for n = k
is divisible by 6. We can write the statement as
In this step, we need to prove that if the statement is true for n = k, then it is also true for n = k + 1.
Now, the term k(k+1) is always even. You can refer here for the proof.
So, we can write
Substituting the above equation in (2), we get
is divisible by 6. Thus, the given statement is true for n = k + 1 if it is true for n = k. Therefore, by the principle of mathematic induction, the given statement “n3 + 5n is divisible by 6″ is true ∀ n ∈ N.