As indicated by the inquiry,

\[~P\left( n \right)\text{ }=\text{ }7n\text{ }\text{ }2n\] is distinct by 5.

In this way, subbing various qualities for n, we get,

\[~P\left( 0 \right)\text{ }=\text{ }70\text{ }\text{ }20\text{ }=\text{ }0\] Which is distinguishable by 5.

\[~P\left( 1 \right)\text{ }=\text{ }71\text{ }\text{ }21\text{ }=\text{ }5\] Which is distinguishable by 5.

\[~P\left( 2 \right)\text{ }=\text{ }72\text{ }\text{ }22\text{ }=\text{ }45\] Which is distinguishable by 5.

\[~P\left( 3 \right)\text{ }=\text{ }73\text{ }\text{ }23\text{ }=\text{ }335\] Which is distinguishable by 5.

Let \[P\left( k \right)\text{ }=\text{ }7k\text{ }\text{ }2k\] be distinguishable by 5

Thus, we get,

\[\begin{array}{*{35}{l}}

~  \\

\Rightarrow 7k2k\text{ }=\text{ }5x.  \\

\end{array}\]

Presently, we additionally get that,

\[\begin{array}{*{35}{l}}

~  \\

\Rightarrow P\left( k+1 \right)=\text{ }7k+12k+1  \\

\end{array}\]

\[=\text{ }\left( 5\text{ }+\text{ }2 \right)7k\text{ }\text{ }2\left( 2k \right)\]

\[=\text{ }5\left( 7k \right)\text{ }+\text{ }2\text{ }\left( 7k\text{ }\text{ }2k \right)\]

\[=\text{ }5\left( 7k \right)\text{ }+\text{ }2\text{ }\left( 5x \right)\] Which is distinguishable by 5.

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

In this manner, by Mathematical Induction,

\[P(n)={{7}^{n}}~\text{ }{{2}^{n}}~\] is separable by 5 is valid for every regular number n.