As per the inquiry,
![\[P\left( n \right)\text{ }=\text{ }n\left( n^2\text{ }+\text{ }5 \right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-8b7b76aad8a2d38e9b894e473376becc_l3.png "Rendered by QuickLaTeX.com")
is distinguishable by 6.
Along these lines, subbing various qualities for n, we get,
![\[P\left( 0 \right)\text{ }=\text{ }0\left( 0^2\text{ }+\text{ }5 \right)\text{ }=\text{ }0\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-2e32d4404ed187416bbf7c058c7f9d8b_l3.png "Rendered by QuickLaTeX.com")
Which is separable by 6.
![\[P\left( 1 \right)\text{ }=\text{ }1\left( 1^2\text{ }+\text{ }5 \right)\text{ }=\text{ }6\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-ed436cb8b1f0ae424fabb687ed80b2f9_l3.png "Rendered by QuickLaTeX.com")
Which is separable by 6.
![\[P\left( 2 \right)\text{ }=\text{ }2\left( 2^2\text{ }+\text{ }5 \right)\text{ }=\text{ }18\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-735b38919b34469ab2f89bd1c1025b49_l3.png "Rendered by QuickLaTeX.com")
Which is separable by 6.
![\[P\left( 3 \right)\text{ }=\text{ }3\left( 3^2\text{ }+\text{ }5 \right)\text{ }=\text{ }42\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-bcb93c89d16c117a05db982f3ad828af_l3.png "Rendered by QuickLaTeX.com")
Which is separable by 6.
Let
![\[P\left( k \right)\text{ }=\text{ }k\left( k^2\text{ }+\text{ }5 \right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-ca89ce2a472082b2540170163daa2e18_l3.png "Rendered by QuickLaTeX.com")
be separable by 6.
Thus, we get,
![\[\Rightarrow k\left( k^2\text{ }+\text{ }5 \right)\text{ }=\text{ }6x.\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-b6fab266188c23f3ed9f8d180945d1bb_l3.png "Rendered by QuickLaTeX.com")
Presently, we additionally get that,
![\[\Rightarrow P\left( k+1 \right)\text{ }=\text{ }\left( k+1 \right)\left( \left( k+1 \right)^2\text{ }+\text{ }5 \right)\text{ }=\text{ }\left( k+1 \right)\left( k^2+2k+6 \right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-ce1c05aea11298c5669b321c55f40205_l3.png "Rendered by QuickLaTeX.com")
![\[=\text{ }k^3\text{ }+\text{ }3k^2\text{ }+\text{ }8k\text{ }+\text{ }6\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-c591aaf356a1e68910df7397685d1442_l3.png "Rendered by QuickLaTeX.com")
![\[=\text{ }6x+3k^2+3k+6\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-85095c715527e2f8d289b6c692c28390_l3.png "Rendered by QuickLaTeX.com")
![\[=\text{ }6x+3k\left( k+1 \right)+6[n\left( n+1 \right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-980793bb74ed106703e76a4647ca8229_l3.png "Rendered by QuickLaTeX.com")
is in every case even and separable by 2]
![\[=\text{ }6x\text{ }+\text{ }3\times 2y\text{ }+\text{ }6\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-57fa8b658c282412d00ff43a3497f1e0_l3.png "Rendered by QuickLaTeX.com")
Which is detachable by 6.
![\[\Rightarrow P\left( k+1 \right)\]](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.
In this manner, by Mathematical Induction,
is separable by 6, for every normal number n.