CAT 2021 Slot 2 QA Question 5: For all possible integers $n$ satisfying $2.25 \leq 2+2^{n+2} \leq 202$, the number of integer values of $3+3^{n+1}$ is

Question

For all possible integers $n$ satisfying $2.25 \leq 2+2^{n+2} \leq 202$, the number of integer values of $3+3^{n+1}$ is

Accepted Answer

Start with the given inequality.

$2.25 \leq 2+2^{n+2} \leq 202$
$0.25 \leq 2^{n+2} \leq 200$

Now, $0.25 = 2^{-2}$ and $2^7 = 128 \leq 200

$-2 \leq n+2 \leq 7$
$-4 \leq n \leq 5$

So $n$ can take the 10 integer values $-4, -3, -2, -1, 0, 1, 2, 3, 4$ and $5$.

For $3 + 3^{n+1}$ to be an integer, $

For all possible integers nnn satisfying 2.25≤2+2n+2≤2022.25 \leq 2+2^{n+2} \leq 2022.25≤2+2n+2≤202, the number of integer values of 3+3n+13+3^{n+1}3+3n+1 is