Question

For all possible integers n satisfying 2.25 ≤ 2 + 2n + 2 ≤ 202, the number of integer values of 3 + 3n + 1 is

Answer 1

Correct Answer: 7

The inequality 2.25 ≤ 2 + 2n + 2 ≤ 202 can be simplified to ensure that 2 + 2n + 2 is less than or equal to 202.

If we take 27 = 128, this satisfies the inequality, but 28 does not.

So, the maximum value for n is 5.

Additionally, when subtracting 2 from both sides of the inequality, we get 0.25 ≤ 2n + 2, which can be further simplified to 2 - 2 ≤ 2n + 2. This implies that n ≥ -4.

Therefore, the integral values of n can range from -4 to 5.

For the expression 3 + 3n + 1 to have integral values, n must be greater than or equal to -1 and go up to 5. Hence, the possible values of n are -1, 0, 1, 2, 3, 4, and 5.

So, there are 7 possible values for n.