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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