Question 12 Marks

If $P(n)$ is the statement "$n^3 + n$ is divisible by $3$", prove that $P(3)$ is true but $P(4)$ is not true.

Answer

$P(n): n^3+n$ is divisible by $3$
$P (3)$ : $3^3+3$ is divisible by $3$
$\Rightarrow P(3): 30$ is divisible by $3$
$\therefore P (3)$ is true.
Now,
$P(4): 4^3+3=67$ is divisible by $3$
Since, $67$ is not divisible by $3$
So, $P (4)$ is not true.

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