M2 Mathematical Induction

Let P(n) Be the statement n^3 + 5n is divisible by 6.

When n = 1, n^3 + 5n = 6 which is divisible by 6.

So P(1) is true.

Suppose that P(k) is true for some positive integer k, then

k^3 + 5k = 6M where M is a positive integer.

When n = k + 1:

(k + 1)^3 + 5(k + 1) = k^3 + 3k^2 + 8k + 6

= (k^3 + 5k) + 3(k^2 + k + 2)

= 6M + 3(k^2 + k + 2)

When k is odd, k^2 is odd and so k^2 + k + 2 is even

When k is even, k^2 is even and so k^2 + k + 2 is even

Hence 3(k^2 + k + 2) is divisible by 6 and so P(k + 1) is also true.

By the first principle of MI, P(n) is true for all positive integers n.