We can compose the given assertion as

\[P\left( n \right):\text{ }\left( 2n\text{ }+7 \right)\text{ }<\text{ }\left( n\text{ }+\text{ }3 \right)2\]

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

$2.1+7=9<{{(1+3)}^{2}}=16$

Which is valid.

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

$(2k+7)<{{(k+3)}^{2}}\text{ n}(1)$

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

Here

\[\left\{ 2\text{ }\left( k\text{ }+\text{ }1 \right)\text{ }+\text{ }7 \right\}\text{ }=\text{ }\left( 2k\text{ }+\text{ }7 \right)\text{ }+\text{ }2\]

We can compose it as

\[=\text{ }\left\{ 2\text{ }\left( k\text{ }+\text{ }1 \right)\text{ }+\text{ }7 \right\}\]

From condition (1) we get

$(2k+7)+2<{{(k+3)}^{2}}+2$

By growing the terms

$2(k+1)+7<{{k}^{2}}+6k+9+2$

On additional estimation

$2(k+1)+7<{{k}^{2}}+6k+11$

Here ${{k}^{2}}+6k+11<{{k}^{2}}+8k+16$

We can compose it as

$\begin{align}

& 2(k+1)+7<{{(k+4)}^{2}} \\

& 2(k+1)+7<(k+1)+{{3}^{2}} \\

\end{align}$

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

Consequently, by the guideline of numerical enlistment, proclamation \[P\text{ }\left( n \right)\]is valid for all normal numbers for example \[n.\]