Question

Prove the following statements with either induction, strong induction or proof by smallest counterexample. Prove that \(3 \mid\left(n^{3}+5 n+6\right)\) for every integer \(n \geq 0\).

Short answer

The statement \(3 \mid\left(n^{3}+5 n+6\right)\) is true for every integer \(n \geq 0\) based on the principles of mathematical induction.

Step-by-step solution

Base Case

Verify the statement for the base case i.e., \(n=0\). Substitute \(n=0\) in \(n^{3}+5 n+6\), the result is 6, which is divisible by 3. Therefore, the statement holds for the base case.

Induction Hypothesis

Now, we assume that the given statement is true for an arbitrary positive integer \(k\), that is, \(3 \mid\left(k^{3}+5k+6\right)\).

Induction Step

We now need to show that the statement is true for \(k+1\), under the assumption that it holds for \(k\). So we substitute \(n = k+1\) in \(n^{3}+5 n+6\) obtaining \((k+1)^{3}+5(k+1)+6\). After simplifying, we get \(k^{3}+3k^{2}+3k+1+5k+5+6\), which simplifies to \(k^{3}+5k+6+3k^{2}+8k+6\). According to our induction hypothesis, \(3 \mid\left(k^{3}+5k+6\right)\), so we can say that the first part of the expression is divisible by 3. Given that the second part, \(3k^{2}+8k+6\), is also divisible by 3, we can deduce that the statement holds for \(k+1\) when it's true for \(k\).