We can compose the given assertion as
\[\mathbf{P}\text{ }\left( \mathbf{n} \right):\text{ }\mathbf{1}.\mathbf{2}\text{ }+\text{ }\mathbf{2}.\mathbf{22}\text{ }+\text{ }\mathbf{3}.\mathbf{22}\text{ }+\text{ }\ldots \text{ }+\text{ }\mathbf{n}.\mathbf{2n}\text{ }=\text{ }\left( \mathbf{n}\text{ }\text{ }\mathbf{1} \right)\text{ }\mathbf{2n}+\mathbf{1}\text{ }+\text{ }\mathbf{2}\]
In the event that \[\mathbf{n}\text{ }=\text{ }\mathbf{1}\]we get
\[\mathbf{P}\text{ }\left( \mathbf{1} \right):\text{ }\mathbf{1}.\mathbf{2}\text{ }=\text{ }\mathbf{2}\text{ }=\text{ }\left( \mathbf{1}\text{ }\text{ }\mathbf{1} \right)\text{ }\mathbf{21}+\mathbf{1}\text{ }+\text{ }\mathbf{2}\text{ }=\text{ }\mathbf{0}\text{ }+\text{ }\mathbf{2}\text{ }=\text{ }\mathbf{2}\]
Which is valid.
Think about \[\mathbf{P}\text{ }\left( \mathbf{k} \right)\]be valid for some sure number \[\mathbf{k}\]
\[\mathbf{1}.\mathbf{2}\text{ }+\text{ }\mathbf{2}.\mathbf{22}\text{ }+\text{ }\mathbf{3}.\mathbf{22}\text{ }+\text{ }\ldots \text{ }+\text{ }\mathbf{k}.\mathbf{2k}\text{ }=\text{ }\left( \mathbf{k}\text{ }\text{ }\mathbf{1} \right)\text{ }\mathbf{2k}\text{ }+\text{ }\mathbf{1}\text{ }+\text{ }\mathbf{2}\text{ }\ldots \text{ }\left( \mathbf{I} \right)\]
Presently let us demonstrate that \[\mathbf{P}\text{ }\left( \mathbf{k}\text{ }+\text{ }\mathbf{1} \right)\]is valid.
Here
\[P\text{ }\left( k\text{ }+\text{ }1 \right)\]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 \[n.\]