Number of integer values of expression

Given inequality: $2.25 \le 2 + 2^{n + 2} \le 202$ $0.25 \le 2^{n + 2} \le 200$ $2^{-2} \le 2^{n + 2} \le 200$

Taking logarithm base 2: $-2 \le n + 2 \le \log_2 200$ Since $2^7 = 128 < 200 < 256 = 2^8$, we have $7 < \log_2 200 < 8$.

For $n$ to be an integer: $-4 \le n \le 5$

So $n \in \{-4, -3, -2, -1, 0, 1, 2, 3, 4, 5\}$.

We need $3 + 3^{n + 1}$ to be an integer. $3 + 3^{n + 1}$ is an integer if $n + 1 \ge 0 \implies n \ge -1$.

Integer values of $n \ge -1$ are $n \in \{-1, 0, 1, 2, 3, 4, 5\}$, which gives 7 values.