If \[P\left( n \right)\] stands for the statement “$n\left( n+1 \right)\left( n+2 \right)$ is divisible by 6”, then what is $P\left( 3 \right)$?

Hint: We will look at what \[P\left( n \right)\] signifies, where $n$ is a natural number. We will see how we can substitute different values of $n$ and check if the statement denoted by $P\left( n \right)$ holds true for these values of $n$. After that we will look at the statement after substituting $n=3$. This will tell us the statement $P\left( 3 \right)$.

Complete step by step answer:
\[P\left( n \right)\] stands for the statement “$n\left( n+1 \right)\left( n+2 \right)$ is divisible by 6”. Here, $n$ is a natural number. We will substitute $n=1$ to find the truth value of the statement represented by $P\left( 1 \right)$. So, for $n=1$ we have,
$P\left( 1 \right):1\left( 1+1 \right)\left( 1+2 \right)\text{ is divisible by 6}$.
We know that $1\left( 1+1 \right)\left( 1+2 \right)=2\times 3=6$ which is divisible by 6. So, $P\left( 1 \right)$ is true. Let us substitute $n=2$ and check if the statement is true. For $n=2$, we have
$P\left( 2 \right):2\left( 2+1 \right)\left( 2+2 \right)\text{ is divisible by 6}$.
We know that $2\left( 2+1 \right)\left( 2+2 \right)=2\times 3\times 4=24$ and 24 is divisible by 6. Next we will substitute $n=3$, we get the following,
$P\left( 3 \right):3\left( 3+1 \right)\left( 3+2 \right)\text{ is divisible by 6}$.

So, we have obtained the statement $P\left( 3 \right)$ and we can see that $3\left( 3+1 \right)\left( 3+2 \right)=3\times 4\times 5=60$ which is divisible by 6.

Note: This form of representing a statement is used in the principle of mathematical induction. The principle is stated as follows: Suppose there is a given statement $P\left( n \right)$ involving the natural number $n$ such that (i) The statement is true for $n=1$, i.e., $P\left( 1 \right)$ is true, and (ii) If the statement is true for $n=k$ (where $k$ is some positive integer), then the statement is also true for $n=k+1$, i.e., truth of $P\left( k \right)$ implies the truth of $P\left( k+1 \right)$. Then, $P\left( n \right)$ is true for all natural numbers $n$. We can use this principle for the given statement and prove that it is true for all the natural numbers.