SOLUTION:-

By additional improvement

\[=\text{ }\left( \mathbf{k}\text{ }+\text{ }\mathbf{1} \right)\text{ }+\text{ }\mathbf{1}\]

\[P\text{ }\left( k\text{ }+\text{ }1 \right)\]is valid at whatever point \[P\text{ }\left( k \right)\]is valid.

Thus, by the rule of numerical enlistment, articulation \[P\text{ }\left( n \right)\]is valid for all regular numbers for example \[n.\]