As indicated by the inquiry,

\[P\left( n \right)\text{ }=\text{ }32n\text{ }\text{ }1\] is distinct by 8.

Along these lines, subbing various qualities for n, we get,

\[P\left( 0 \right)\text{ }=\text{ }30\text{ }\text{ }1\text{ }=\text{ }0\] which is separable by 8.

\[P\left( 1 \right)\text{ }=\text{ }32\text{ }\text{ }1\text{ }=\text{ }8\] which is distinguishable by 8.

\[P\left( 2 \right)\text{ }=\text{ }34\text{ }\text{ }1\text{ }=\text{ }80\] which is detachable by 8.

\[P\left( 3 \right)\text{ }=\text{ }36\text{ }\text{ }1\text{ }=\text{ }728\] which is distinguishable by 8.

Let \[P\left( k \right)\text{ }=\text{ }32k\text{ }\text{ }1\] be detachable by 8

In this way, we get,

\[\Rightarrow 32k1\text{ }=\text{ }8x.\]

Presently, we likewise get that,

\[\Rightarrow P\left( k+1 \right)\text{ }=\text{ }32\left( k+1 \right)1\]

\[=\text{ }32\left( 8x\text{ }+\text{ }1 \right)\text{ }\text{ }1\]

\[=\text{ }72x\text{ }+\text{ }8\] is distinct by 8.

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

Subsequently, by Mathematical Induction,

\[P\left( n \right)\text{ }=\text{ }32n\text{ }\text{ }1\] is detachable by 8, for all regular numbers n.