Answer to Question #198635 in Calculus for Moe

Question

3) Prove by induction that for all natural numbers n:

(f) n³ + 2n is divisible by 3.

Answer 1

Let us prove by induction that $n^3 + 2n$ is divisible by 3 for all natural numbers $n$.

For $n=1$ we have that $1^3+2\cdot 1=3$ is divisible by 3.

Suppose that for $n=k$ we have that $k^3 + 2k$ is divisible by 3.

Let us prove the statement for $n=k+1.$

Taking into account that

$(k+1)^3+2(k+1)=k^3+3k^2+3k+1+2k+2=(k^3+2k)+3(k^2+k+1)$

and $k^3+2k$ and $3(k^2+k+1)$ are divisible by 3, we conclude that $(k+1)^3+2(k+1)$ is also divisible by 3.

We conclude that $n^3 + 2n$ is divisible by 3 for all natural numbers $n$.