We can compose the given assertion as

$P(n):{{3}^{2}}n+2-8n-9$is separable by \[8\]

In the event that \[n\text{ }=\text{ }1\]we get

$P(1)={{3}^{(2\text{  }\!\!\times\!\!\text{ }1)}}+2-8\text{  }\!\!\times\!\!\text{ }1-9=64$, which is distinguishable by \[8\]

Which is valid.

Think about \[P\text{ }\left( k \right)\]be valid for some sure number \[k\]

${{3}^{2}}k+2-8k-9$is distinct by \[8\]

${{3}^{2}}k+2-8k-9=8m,wherem?N\text{ n}(1)$

Presently let us demonstrate that \[P\text{ }\left( k\text{ }+\text{ }1 \right)\]is valid.

Here

${{3}^{(2(}}k+1)+2)-8(k+1)-9$

We can compose it as

$={{3}^{(2k+2)}}{{.3}^{2}}-8k-8-9$

By adding and taking away \[8k\text{ }and\text{ }9\] we get

$={{3}^{2}}({{3}^{(2k)}}+2-8k-9+8k+9)-8k-17$

On additional improvement

$={{3}^{2}}({{3}^{(2k)}}+2-8k-9)+{{3}^{2}}(8k+9)-8k-17$

From condition \[\left( 1 \right)\]we get

\[=\text{ }9.\text{ }8m\text{ }+\text{ }9\text{ }\left( 8k\text{ }+\text{ }9 \right)\text{ }\text{ }8k\text{ }\text{ }17\]

By duplicating the terms

\[=\text{ }9.\text{ }8m\text{ }+\text{ }72k\text{ }+\text{ }81\text{ }\text{ }8k\text{ }\text{ }17\]

So we get

\[=\text{ }9.\text{ }8m\text{ }+\text{ }64k\text{ }+\text{ }64\]

By taking out the normal terms

\[=\text{ }8\text{ }\left( 9m\text{ }+\text{ }8k\text{ }+\text{ }8 \right)\]

\[=\text{ }8r\], where \[r\text{ }=\text{ }\left( 9m\text{ }+\text{ }8k\text{ }+\text{ }8 \right)\]is a characteristic number

So ${{3}^{(2(}}k+1)+2)-8(k+1)-9$ is distinct by \[8\]

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

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