Given that, $2.25 \leq 2 + 2^{n+2} \leq 202 $

$ \Rightarrow 0.25 \leq 2^{n+2} \leq 200 $

$ \Rightarrow 2^{-2} \leq 2^{n+2} < 2^{8} \quad[{\color {Blue}{\because 2^{7} = 128, 2^{8} = 256 \Rightarrow 2^{7} < 200 < 2^{8}}}]$

$ \Rightarrow\; – 2 \leq n+2 < 8 $

$ \Rightarrow \;– 4 \leq n < 6 $

$ \Rightarrow n \in \{ – 4 , – 3, – 2, – 1, 0, 1, 2, 3, 4, 5 \} $

The integer value of $3 + 3^{n+1} :$

$\Rightarrow n + 1 \geq 0 $

$\Rightarrow \boxed{ n \geq\; – 1} $

$\Rightarrow n \in \{– 1, 0, 1, 2, 3, 4, 5 \} $

$\therefore$ The number of integer values of $3 + 3^{n+1}$ is $7.$

Correct Answer $: 7$