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 \(2.25 \leq 2 + 2n + 2 \leq 202\) 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.

Step 1: Simplifying the Inequality

We begin by subtracting 2 from both sides of the inequality: \[ 2.25 \leq 2 + 2n + 2 \leq 202 \quad \Rightarrow \quad 0.25 \leq 2n + 2 \leq 200. \]

Now, subtracting 2 from all parts of the inequality: \[ 0.25 - 2 \leq 2n + 2 - 2 \leq 200 - 2 \quad \Rightarrow \quad -1.75 \leq 2n \leq 198. \]

Step 2: Solving for \(n\)

Next, divide the entire inequality by 2: \[ \frac{-1.75}{2} \leq n \leq \frac{198}{2} \quad \Rightarrow \quad -0.875 \leq n \leq 99. \]

Since \(n\) must be an integer, we round \(n\) to the nearest integer within this range: \[ n \geq 0 \quad \text{and} \quad n \leq 5. \]

Step 3: Calculating the Integral Values of \(n\)

Thus, the possible integral values of \(n\) range from -4 to 5.

Step 4: Ensuring that \(n\) is greater than or equal to -1

For the expression \(3 + 3n + 1\) to have integral values, \(n\) must be greater than or equal to -1 and go up to 5.

The possible values of \(n\) are: \[ n = -1, 0, 1, 2, 3, 4, 5. \]

Conclusion

Thus, there are 7 possible values for \(n\): -1, 0, 1, 2, 3, 4, and 5.