As indicated by the inquiry,
![\[P\left( n \right)\text{ }=\text{ }32n\text{ }\text{ }1\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-a61f1f5c71002907848fed4360b787e6_l3.png "Rendered by QuickLaTeX.com")
is distinct by 8.
Along these lines, subbing various qualities for n, we get,
![\[P\left( 0 \right)\text{ }=\text{ }30\text{ }\text{ }1\text{ }=\text{ }0\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-72f321958b4baea5a9db7ab9287b4d4e_l3.png "Rendered by QuickLaTeX.com")
which is separable by 8.
![\[P\left( 1 \right)\text{ }=\text{ }32\text{ }\text{ }1\text{ }=\text{ }8\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-72e1479ae0e620c2fbaf153832bf3db7_l3.png "Rendered by QuickLaTeX.com")
which is distinguishable by 8.
![\[P\left( 2 \right)\text{ }=\text{ }34\text{ }\text{ }1\text{ }=\text{ }80\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-63191eb03e02faf9f962c9420a08512b_l3.png "Rendered by QuickLaTeX.com")
which is detachable by 8.
![\[P\left( 3 \right)\text{ }=\text{ }36\text{ }\text{ }1\text{ }=\text{ }728\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-266ce65fe60eb25cd1ec4252296569f6_l3.png "Rendered by QuickLaTeX.com")
which is distinguishable by 8.
Let
![\[P\left( k \right)\text{ }=\text{ }32k\text{ }\text{ }1\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-e3896e6e4763f59e1f522e811d6c795e_l3.png "Rendered by QuickLaTeX.com")
be detachable by 8
In this way, we get,
![\[\Rightarrow 32k1\text{ }=\text{ }8x.\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-19550cae8fb7603e0f898e1a5998281f_l3.png "Rendered by QuickLaTeX.com")
Presently, we likewise get that,
![\[\Rightarrow P\left( k+1 \right)\text{ }=\text{ }32\left( k+1 \right)1\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-af8144af2a8fd2c46eb1c3801dbbc06b_l3.png "Rendered by QuickLaTeX.com")
![\[=\text{ }32\left( 8x\text{ }+\text{ }1 \right)\text{ }\text{ }1\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-947cfd5f53ba99b7731b11580a01df7b_l3.png "Rendered by QuickLaTeX.com")
![\[=\text{ }72x\text{ }+\text{ }8\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-39282516cb1e4d8a77b5f50ab1229cc5_l3.png "Rendered by QuickLaTeX.com")
is distinct by 8.
![\[\Rightarrow P\left( k+1 \right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-0d17e4216b59f77dd0be0d0245938e26_l3.png "Rendered by QuickLaTeX.com")
is valid when P(k) is valid.
Subsequently, by Mathematical Induction,
is detachable by 8, for all regular numbers n.