As per the inquiry,

\[P\left( n \right)\text{ }=\text{ }4n\text{ }\text{ }1\] is separable by 3.

Thus, subbing various qualities for n, we get,

\[P\left( 0 \right)\text{ }=\text{ }40\text{ }\text{ }1\text{ }=\text{ }0\] which is distinguishable by 3.

\[P\left( 1 \right)\text{ }=\text{ }41\text{ }\text{ }1\text{ }=\text{ }3\] which is detachable by 3.

\[P\left( 2 \right)\text{ }=\text{ }42\text{ }\text{ }1\text{ }=\text{ }15\] which is distinct by 3.

\[P\left( 3 \right)\text{ }=\text{ }43\text{ }\text{ }1\text{ }=\text{ }63\]which is distinct by 3.

Let \[P\left( k \right)\text{ }=\text{ }4k\text{ }\text{ }1\] be detachable by 3,

In this way, we get,

\[\Rightarrow 4k1\text{ }=\text{ }3x\] .

Presently, we likewise get that,

\[\Rightarrow P\left( k+1 \right)\text{ }=\text{ }4k+11\]

\[=\text{ }4\left( 3x\text{ }+\text{ }1 \right)\text{ }\text{ }1\]

\[=\text{ }12x\text{ }+\text{ }3\] is detachable by 3.

\[\Rightarrow P\left( k+1 \right)\] is valid when P(k) is valid

Hence, by Mathematical Induction,

\[P\left( n \right)\text{ }=\text{ }4n\text{ }\text{ }1\] is detachable by 3 is valid for every regular number n.