I need to prove: n^{3 }+ 5n is divisible by 6.
n=1
1^{3}+5(1)=6 which is divisible by 6 = true.
n=2
2^{3}+5(2)=18 which is divisible by 6 = true.
What else do I need to prove?
Suppose that $\displaystyle K^3+5K$ is divisble by six, where $\displaystyle K\ge 3$ is a positive integer.
Using that fact, prove that
$\displaystyle \begin{align*} (K+1)^3+5(K+1)&=(K^3+3K^2+3K+1)+(5K+5)\\&=(K^3+5K) +3K(3K+1)+6\end{align*}$
is divisble by six.
HINT: The sum of three multiples of six is divisble by six.
Remember that steps to induction goes like this
Let P be a statement. Show that
P(1) is true
If P(k) is true, then P(k+1) is true
You already showed that P(1) is true. Now just need to show P(k+1) is true if P(k) is true. Let P(k) be true. Then k^3 + 5k is divisible by 6. Need to show that (k+1)^3 + 5(k+1) is divisible by 6. Expand.
(k+1)^3 + 5(k+1) = k^3 + 3k^2 + 3k + 1 + 5k + 5 = (k^3 + 5k) + 3k(k+1) + 6.
Clearly 6 is divisible by 6. By induction assumption k^3 + 5k is divisible by 6. I claim that 3k(k+1) is divisible by 6. That is because k(k+1) is even for all k. If k is even, then clearly k(k+1) is even so 3k(k+1) is divisible by 6. If k is odd, then k + 1 is even so k(k+1) is even and again 3k(k+1) is divisible by 6. The sum of terms each of which are divisible by 6 is also divisible by 6.
I know the OP probably needed to use induction. For completeness, I'll give a proof without induction. I hope this gives more intuition as to why the statement is true.
Lemma
$\displaystyle n^3+5n$ is divisible by 6 if and only if $\displaystyle n^3-n$ is divisible by 6.
Proof: Suppose $\displaystyle n^3+5n=6k$, then $\displaystyle n^3-n=6(k-n)$. $\displaystyle k>n$ because $\displaystyle k= \frac{n(n^2+5)}{6}\geq n$ since $\displaystyle n^2+5 \geq 6$ for $\displaystyle n\in \mathbb{N}$. Conversely, $\displaystyle n^3-n=6k \Rightarrow n^3+5n=6(k+n)$.
Proposition
$\displaystyle n^3+5n$ is divisible by 6.
Proof: From the lemma, the proposition holds iff $\displaystyle n^3-n$ is divisible by 6. But $\displaystyle n^3-n=n(n^2-1)=(n-1)(n)(n+1)$. These are three consecutive integers, and so one of them must be divisible by 3.